Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability.

Caratheodory’s definition of mea

Math212a1411
Lebesgue measure.

Shlomo Sternberg

October 14, 2014

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Reminder

No class this Thursday

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Today’s lecture will be devoted to Lebesgue measure, a creation of

Henri Lebesgue, in his thesis, one of the most famous theses in the
history of mathematics.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Henri Léon Lebesgue

Born: 28 June 1875 in Beauvais, Oise, Picardie, France

Died: 26 July 1941 in Paris, France

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

In today’s lecture we will discuss the concept of measurability of a

subset of R. We will begin with Lebesgue’s (1902) definition of
measurability, which is easy to understand and intuitive. We will
then give Caratheodory’s (1914) definition of measurabiity which is
highly non-intuitive but has great technical advantage. For subsets
of R these two definitions are equivalent (as we shall prove). But
the Caratheodory definition extends to many much more general
situations.
In particular, the Caratheodory definition will prove useful for us
later, when we study Hausdorff measures.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

The /2n trick.

An argument which will recur several times is:

∞
X 1
=1
2n
1

and so
∞
X 
= .
2n
1

We will call this the “/2n trick” and not spell it out every time we
use it.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

1 Lebesgue outer measure.

2 Lebesgue inner measure.

3 Lebesgue’s definition of measurability.

4 Caratheodory’s definition of measurability.

5 Countable additivity.

6 σ-fields, measures, and outer measures.

7 The Borel-Cantelli lemmas

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

The definition of Lebesgue outer measure.

For any subset A ⊂ R we define its Lebesgue outer measure by

X [
m∗ (A) := inf `(In ) : In are intervals with A ⊂ In . (1)

Here the length `(I ) of any interval I = [a, b] is b − a with the

same definition for half open intervals (a, b] or [a, b), or open
intervals.
Of course if a = −∞ and b is finite or +∞, or if a is finite and
b = +∞, or if a = −∞ and b = ∞ the length is infinite.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

It doesn’t matter if the intervals are open, half open or

closed.

X [
m∗ (A) := inf `(In ) : In are intervals with A ⊂ In . (1)

The infimum in (1) is taken over all covers of A by intervals. By

the /2n trick, i.e. by replacing each Ij = [aj , bj ] by
(aj − /2j+1 , bj + /2j+1 ) we may assume that the infimum is
taken over open intervals. (Equally well, we could use half open
intervals of the form [a, b), for example.).

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Monotonicity, subadditivity.

If A ⊂ B then m∗ (A) ≤ m∗ (B) since any cover of B by intervals is

a cover of A.
For any two sets A and B we clearly have

m∗ (A ∪ B) ≤ m∗ (A) + m∗ (B).

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Sets of measure zero don’t matter.

A set Z is said to be of (Lebesgue) measure zero it its Lebesgue

outer measure is zero, i.e. if it can be covered by a countable
union of (open) intervals whose total length can be made as small
as we like.
If Z is any set of measure zero, then m∗ (A ∪ Z ) = m∗ (A).

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

The outer measure of a finite interval is its length.

If A = [a, b] is an interval, then we can cover it by itself, so

m∗ ([a, b]) ≤ b − a,

and hence the same is true for (a, b], [a, b), or (a, b). If the interval
is infinite, it clearly can not be covered by a set of intervals whose
total length is finite, since if we lined them up with end points
touching they could not cover an infinite interval. We claim that

m∗ (I ) = `(I ) (2)
if I is a finite interval.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Proof.

We may assume that I = [c, d] is a closed interval by what we

have already said, and that the minimization in (1) is with respect
to a cover by open intervals. So what we must show is that if
[
[c, d] ⊂ (ai , bi )
i

then X
d −c ≤ (bi − ai ).
i

We first apply Heine-Borel to replace the countable cover by a

finite cover. (This only decreases the right hand side of preceding
inequality.) So let n be the number of elements in the cover.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

We need to prove that if

n
[ n
X
[c, d] ⊂ (ai , bi ) then d − c ≤ (bi − ai ).
i=1 i=1

We do this this by induction on n. If n = 1 then a1 < c and

b1 > d so clearly b1 − a1 > d − c.
Suppose that n ≥ 2 and we know the result for all covers (of all
intervals [c, d] ) with at most n − 1 intervals in the cover. If some
interval (ai , bi ) is disjoint from [c, d] we may eliminate it from the
cover, and then we are in the case of n − 1 intervals. So we may
assume that every (ai , bi ) has non-empty intersection with [c, d].

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Among the the intervals (ai , bi ) there will be one for which ai takes
on the minimum possible value. By relabeling, we may assume that
this is (a1 , b1 ). Since c is covered, we must have a1 < c. If b1 > d
then (a1 , b1 ) covers [c, d] and there is nothing further to do. So
assume b1 ≤ d. We must have b1 > c since (a1 , b1 ) ∩ [c, d] 6= ∅.
Since b1 ∈ [c, d], at least one of the intervals (ai , bi ), i > 1
contains the point b1 . By relabeling, we may assume that it is
(a2 , b2 ). But now we have a cover of [c, d] by n − 1 intervals:
n
[
[c, d] ⊂ (a1 , b2 ) ∪ (ai , bi ).
i=3
Pn
So by induction d − c ≤ (b2 − a1 ) + i=3 (bi − ai ).
But b2 − a1 ≤ (b2 − a2 ) + (b1 − a1 ) since a2 < b1 . 

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

We can use small intervals.

We repeat that the intervals used in (1) could be taken as open,

closed or half open without changing the definition. If we take
them all to be half open, of the form Ii = [ai , bi ), we can write
each Ii as a disjoint union of finite or countably many intervals
each of length < . So it makes no difference to the definition if
we also require the
`(Ii ) <  (3)
in (1). We will see that when we pass to other types of measures
this will make a difference.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Summary of where we are so far.

We have verified, or can easily verify the following properties:
1
m∗ (∅) = 0.
2 A ⊂ B ⇒ m∗ (A) ≤ m∗ (B).
3 By the /2n trick, for any finite or countable union we have
[ X
m∗ ( Ai ) ≤ m∗ (Ai ).
i i

4 If dist (A, B) > 0 then

m∗ (A ∪ B) = m∗ (A) + m∗ (B).
5 m∗ (A) = inf{m∗ (U) : U ⊃ A, U open}.
6 For an interval
m∗ (I ) = `(I ).
Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

The only items that we have not done already are items 4 and 5.
But these are immediate: for 4 we may choose the intervals in (1)
all to have length <  where 2 < dist (A, B) so that there is no
overlap. As for item 5, we know from 2 that m∗ (A) ≤ m∗ (U) for
any set U ⊃ A, in particular for any open set U which contains A.
We must prove the reverse inequality: if m∗ (A) = ∞ this is trivial.
Otherwise, we may take the intervals in (1) to be open and then
the union on the right is an open set whose Lebesgue outer
measure is less than m∗ (A) + δ for any δ > 0 if we choose a close
enough approximation to the infimum.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Lebesgue outer measure on Rn .

All the above works for Rn instead of R if we replace the word

“interval” by “rectangle”, meaning a rectangular parallelepiped, i.e
a set which is a product of one dimensional intervals. We also
replace length by volume (or area in two dimensions). What is
needed is the following:

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Lemma
Let C be a finite non-overlapping collection of closed rectangles all
contained in the closed rectangle J. Then
X
vol J ≥ vol I .
I ∈C

If C is any finite collection of rectangles such that

[
J⊂ I
I ∈C

then X
vol J ≤ vol (I ).
I ∈C

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

This lemma occurs on page 1 of Stroock, A concise introduction to

the theory of integration together with its proof. I will take this for
granted. In the next few slides I will talk as if we are in R, but
everything goes through unchanged if R is replaced by Rn .
In fact, once we will have developed enough abstract theory, we
will not need this lemma. So for those of you who are purists and
do not want to go through the lemma in Stroock, just stick to R.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

The definition of Lebesgue inner measure.

Item 5. in our list said that the Lebesgue outer measure of any set
is obtained by approximating it from the outside by open sets. The
Lebesgue inner measure is defined as

m∗ (A) = sup{m∗ (K ) : K ⊂ A, K compact }. (4)

Clearly
m∗ (A) ≤ m∗ (A)
since m∗ (K ) ≤ m∗ (A) for any K ⊂ A. We also have

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

The Lebesgue inner measure of an interval.

Proposition.
For any interval I we have

m∗ (I ) = `(I ). (5)

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Proof.
If `(I ) = ∞ the result is obvious. So we may assume that I is a
finite interval which we may assume to be open, I = (a, b). If
K ⊂ I is compact, then I is a cover of K and hence from the
definition of outer measure m∗ (K ) ≤ `(I ). So m∗ (I ) ≤ `(I ). On
the other hand, for any  > 0,  < 12 (b − a) the interval
[a + , b − ] is compact and
m∗ ([a − , a + ]) = b − a − 2 ≤ m∗ (I ). Letting  → 0 proves the
proposition.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Lebesgue’s definition of measurability for sets of finite

outer measure.

A set A with m∗ (A) < ∞ is said to measurable in the sense of

Lebesgue if
m∗ (A) = m∗ (A). (6)
If A is measurable in the sense of Lebesgue, we write

m(A) = m∗ (A) = m∗ (A). (7)

If K is a compact set, then m∗ (K ) = m∗ (K ) since K is a compact

set contained in itself. Hence all compact sets are measurable in
the sense of Lebesgue. If I is a bounded interval, then I is
measurable in the sense of Lebesgue by the preceding Proposition.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Lebesgue’s definition of measurability for sets of infinite

outer measure.

If m∗ (A) = ∞, we say that A is measurable in the sense of

Lebesgue if all of the sets A ∩ [−n, n] are measurable.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Countable additivity.

Theorem
S
If A = Ai is a (finite or) countable disjoint union of sets which
are measurable in the sense of Lebesgue, then A is measurable in
the sense of Lebesgue and
X
m(A) = m(Ai ).
i

In the proof we may assume that m(A) < ∞ - otherwise apply the
result to A ∩ [−n, n] and Ai ∩ [−n, n] for each n.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

We have X X
m∗ (A) ≤ m∗ (An ) = m(An ).
n n

Let  > 0, and for each n choose compact Kn ⊂ An with

 
m∗ (Kn ) ≥ m∗ (An ) − n
= m(An ) − n
2 2
which we can do since An is measurable in the sense of Lebesgue.
The sets Kn are pairwise disjoint, hence, being compact, at
positive distances from one another. Hence

m∗ (K1 ∪ · · · ∪ Kn ) = m∗ (K1 ) + · · · + m∗ (Kn )

and K1 ∪ · · · ∪ Kn is compact and contained in A.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

 
m∗ (Kn ) ≥ m∗ (An ) − = m(An ) − n
2n 2
m∗ (K1 ∪ · · · ∪ Kn ) = m∗ (K1 ) + · · · + m∗ (Kn )
and K1 ∪ · · · ∪ Kn is compact and contained in A. Hence

m∗ (A) ≥ m∗ (K1 ) + · · · + m∗ (Kn ),

and since this is true for all n we have

X
m∗ (A) ≥ m(An ) − .
n

Since this is true for all  > 0 we get

X
m∗ (A) ≥ m(An ).

But then m∗ (A) ≥ m∗ (A) and so they are equal, soPA is

measurable in the sense of Lebesgue, and m(A) = m(Ai ). 
Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Open sets are measurable in the sense of Lebesgue.

Proof.
Any open set O can be written as the countable union of open
intervals Ii , and
n−1
[
Jn := In \ Ii
i=1

is a disjoint collection of intervals (some open, some closed, some

half open) and O is the disjont union of the Jn . So every open set
is a disjoint union of intervals hence measurable in the sense of
Lebesgue.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Closed sets are measurable in the sense of Lebesgue.

This requires a bit more work:

If F is closed, and m∗ (F ) = ∞, then F ∩ [−n, n] is compact, and
so F is measurable in the sense of Lebesgue. Suppose that

m∗ (F ) < ∞.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

For any  > 0 consider the sets

 
G1, := [−1 + ,1 − 2] ∩ F
22 2
 
G2, := ([−2 + 3 , −1] ∩ F ) ∪ ([1, 2 − ] ∩ F)
2 23
 
G3, := ([−3 + 4 , −2] ∩ F ) ∪ ([2, 3 − ] ∩ F)
2 24
..
.

and set [
G := Gi, .
i

The Gi, are all compact, and hence measurable in the sense of
Lebesgue, and the union in the definition of G is disjoint, so G is
measurable in the sense of Lebesgue.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

The Gi, are all compact, and hence measurable in the sense of
Lebesgue, and the union in the definition of G is disjoint, so is
measurable in the sense of Lebesgue. Furthermore, the sum of the
lengths of the “gaps” between the intervals that went into the
definition of the Gi, is . So
X
m(G )+ = m∗ (G )+ ≥ m∗ (F ) ≥ m∗ (G ) = m(G ) = m(Gi, ).
i

In particular, the sum on the right converges, and hence by

considering a finite number of terms, we will have a finite sum
whose value is at least m(G ) − . The corresponding union of sets
will be a compact set K contained in F with

m(K ) ≥ m∗ (F ) − 2.

Hence all closed sets are measurable in the sense of Lebesgue. 

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

The inner-outer characterization of measurability.

Theorem
A is measurable in the sense of Lebesgue if and only if for every
 > 0 there is an open set U ⊃ A and a closed set F ⊂ A such that

m(U \ F ) < .

Proof in one direction. Suppose that A is measurable in the

sense of Lebesgue with m(A) < ∞. Then there is an open set
U ⊃ A with m(U) < m∗ (A) + /2 = m(A) + /2, and there is a
compact set F ⊂ A with m(F ) ≥ m∗ (A) − /2 = m(A) − /2.
Since U \ F is open, it is measurable in the sense of Lebesgue, and
so is F as it is compact. Also F and U \ F are disjoint. Hence
  
m(U \ F ) = m(U) − m(F ) < m(A) + − m(A) − = .
2 2
Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

If A is measurable in the sense of Lebesgue, and m(A) = ∞, we

can apply the above to A ∩ I where I is any compact interval, in
particular to the interval [−n, n]. So there exist open sets
Un ⊃ A ∩ [−n, n] and closed sets
S Fn ⊂ A ∩ [−n, n] with
n
m(Un /Fn ) < /2 . Let U := Un and
[
F := (Fn ∩ ([−n, −n + 1] ∪ [n − 1, n])) .
A convergent sequence in F must eventually lie in an interval of
length one and hence to at most the union of two entries in the
union defining F , so F is closed. We have U ⊃ A ⊃ F and
[
U/F ⊂ (Un /Fn ).
Indeed, any p ∈ U must belong to some ([−n, −n + 1] ∪ [n − 1, n]),
and so if it does not belong to F then it does not belong to Fn . So
X
m(U/F ) ≤ m(Un /Fn ) < . 

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Proof in the other direction.

Suppose that for each , there exist U ⊃ A ⊃ F with
m(U \ F ) < . Suppose that m∗ (A) < ∞. Then m(F ) < ∞ and
m(U) ≤ m(U \ F ) + m(F ) <  + m(F ) < ∞. Then

m∗ (A) ≤ m(U) < m(F ) +  = m∗ (F ) +  ≤ m∗ (A) + .

Since this is true for every  > 0 we conclude that m∗ (A) ≥ m∗ (A)
so they are equal and A is measurable in the sense of Lebesgue.
If m∗ (A) = ∞, we have
U ∩ (−n − , n + ) ⊃ A ∩ [−n, n] ⊃ F ∩ [−n, n] and

m((U ∩ (−n − , n + ) \ (F ∩ [−n, n]) < 2 +  = 3

so we can proceed as before to conclude that

m∗ (A ∩ [−n, n]) = m∗ (A ∩ [−n, n]). 
Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Consequences of the theorem.

Proposition.
If A is measurable in the sense of Lebesgue, so is its complement
Ac = R \ A.

Proof.
Indeed, if F ⊂ A ⊂ U with F closed and U open, then
F c ⊃ Ac ⊃ U c with F c open and U c closed. Furthermore,
F c \ U c = U \ F so if A satisfies the condition of the theorem so
does Ac .

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

More consequences.

Proposiition.
If A and B are measurable in the sense of Lebesgue so is A ∩ B

Proof.
For  > 0 choose UA ⊃ A ⊃ FA and UB ⊃ B ⊃ FB with
m(UA \ FA ) < /2 and m(UB \ FB ) < /2. Then

(UA ∩ UB ) ⊃ (A ∩ B) ⊃ (FA ∩ FB )

and
(UA ∩ UB ) \ (FA ∩ FB ) ⊂ (UA \ FA ) ∪ (UB \ FB ).

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Still more consequences of the theorem.

Proposition.
If A and B are measurable in the sense of Lebesgue then so is
A ∪ B.

Proof.
Indeed, A ∪ B = (Ac ∩ B c )c .

Since A \ B = A ∩ B c we also get

Proposition.
If A and B are measurable in the sense of Lebesgue then so is
A \ B.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Caratheodory’s definition of measurability.

A set E ⊂ R is said to be measurable according to
Caratheodory if for any set A ⊂ R we have

m∗ (A) = m∗ (A ∩ E ) + m∗ (A ∩ E c ) (8)

where (recall that) E c denotes the complement of E . In other

words, A ∩ E c = A \ E . This definition has many advantages, as
we shall see. Our first task will be to show that it is equivalent to
Lebesgue’s.
Notice that we always have m∗ (A) ≤ m∗ (A ∩ E ) + m∗ (A \ E ) so
condition (8) is equivalent to

m∗ (A ∩ E ) + m∗ (A \ E ) ≤ m∗ (A) (9)

for all A.
Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Caratheodory = Lebesgue.

Theorem
A set E is measurable in the sense of Caratheodory if and only if it
is measurable in the sense of Lebesgue.

Proof that Lebesgue ⇒ Caratheodory. Suppose E is

measurable in the sense of Lebesgue. Let  > 0. Choose
U ⊃ E ⊃ F with U open, F closed and m(U/F ) <  which we can
do by the “inner-outer” Theorem. Let V be an open set
containing A. Then A \ E ⊂ V \ F and A ∩ E ⊂ (V ∩ U) so

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

A \ E ⊂ V \ F and A ∩ E ⊂ (V ∩ U) so

m∗ (A \ E ) + m∗ (A ∩ E ) ≤ m(V \ F ) + m(V ∩ U)
≤ m(V \ U) + m(U \ F ) + m(V ∩ U)
≤ m(V ) + .

(We can pass from the second line to the third since both V \ U
and V ∩ U are measurable in the sense of Lebesgue and we can
apply the proposition about disjoint unions.) Taking the infimum
over all open V containing A, the last term becomes m∗ (A) + ,
and as  is arbitrary, we have established (9) showing that E is
measurable in the sense of Caratheodory.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Caratheodory ⇒ Lebesgue, case where m∗ (E ) < ∞.

Then for any  > 0 there exists an open set U ⊃ E with
m(U) < m∗ (E ) + . We may apply condition (8) to A = U to get

m(U) = m∗ (U ∩ E ) + m∗ (U \ E ) = m∗ (E ) + m∗ (U \ E )

so
m∗ (U \ E ) < .
This means that there is an open set V ⊃ (U \ E ) with m(V ) < .
But we know that U \ V is measurable in the sense of Lebesgue,
since U and V are, and

m(U) ≤ m(V ) + m(U \ V )

so
m(U \ V ) > m(U) − .
Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

m(U \ V ) > m(U) − .

So there is a closed set F ⊂ U \ V with m(F ) > m(U) − . But
since V ⊃ U \ E = U ∩ E c , we have

U \ V = U ∩ V c ⊂ U ∩ (U c ∪ E ) = E .

So F ⊂ E . So F ⊂ E ⊂ U and

m(U \ F ) = m(U) − m(F ) < .

Hence E is measurable in the sense of Lebesgue.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Caratheodory ⇒ Lebesgue, case where m∗ (E ) = ∞.

If m(E ) = ∞, we must show that E ∩ [−n, n] is measurable in the

sense of Caratheodory, for then it is measurable in the sense of
Lebesgue from what we already know. We know that the interval
[−n, n] itself, being measurable in the sense of Lebesgue, is
measurable in the sense of Caratheodory. So we will have
completed the proof of the theorem if we show that the intersection
of E with [−n, n] is measurable in the sense of Caratheodory.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Unions and intersections.

More generally, we will show that the union or intersection of two

sets which are measurable in the sense of Caratheodory is again
measurable in the sense of Caratheodory. Notice that the definition
(8) is symmetric in E and E c so if E is measurable in the sense of
Caratheodory so is E c . So it suffices to prove the next lemma to
complete the proof.
Lemma
If E1 and E2 are measurable in the sense of Caratheodory so is
E 1 ∪ E2 .

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Proof of the lemma, 1.

For any set A we have

m∗ (A) = m∗ (A ∩ E1 ) + m∗ (A ∩ E1c )

by (8) applied to E1 . Applying (8) to A ∩ E1c and E2 gives

m∗ (A ∩ E1c ) = m∗ (A ∩ E1c ∩ E2 ) + m∗ (A ∩ E1c ∩ E2c ).

Substituting this back into the preceding equation gives

m∗ (A) = m∗ (A ∩ E1 ) + m∗ (A ∩ E1c ∩ E2 ) + m∗ (A ∩ E1c ∩ E2c ). (10)

Since E1c ∩ E2c = (E1 ∪ E2 )c we can write this as

m∗ (A) = m∗ (A ∩ E1 ) + m∗ (A ∩ E1c ∩ E2 ) + m∗ (A ∩ (E1 ∪ E2 )c ).

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Proof of the lemma, 2.

m∗ (A) = m∗ (A ∩ E1 ) + m∗ (A ∩ E1c ∩ E2 ) + m∗ (A ∩ (E1 ∪ E2 )c ).

Now A ∩ (E1 ∪ E2 ) = (A ∩ E1 ) ∪ (A ∩ (E1c ∩ E2 ) so

m∗ (A ∩ E1 ) + m∗ (A ∩ E1c ∩ E2 ) ≥ m∗ (A ∩ (E1 ∪ E2 )).

Substituting this for the two terms on the right of the previous
displayed equation gives

m∗ (A) ≥ m∗ (A ∩ (E1 ∪ E2 )) + m∗ (A ∩ (E1 ∪ E2 )c )

which is just (9) for the set E1 ∪ E2 . This proves the lemma and
the theorem. 
Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

The class M of measurable sets.

We let M denote the class of measurable subsets of R -

“measurability” in the sense of Lebesgue or Caratheodory these
being equivalent. Notice by induction starting with two terms as in
the lemma, that any finite union of sets in M is again in M

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

The first main theorem in the subject is the following description

of M and the function m on it:
Theorem
M and the function m : M → [0, ∞] have the following
properties:
R ∈ M.
E ∈ M ⇒ E c ∈ M.
S
If En ∈ M for n = 1, 2, 3, . . . then n En ∈ M.
If Fn ∈
SM and the Fn are pairwise disjoint, then
F := n Fn ∈ M and
∞
X
m(F ) = m(Fn ).
n=1

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

We already know the first two items on the list, and we know that
a finite union of sets in M is again in M. We also know the last
assertion. But it will be instructive and useful for us to have a
proof starting directly from Caratheodory’s definition of
measurablity:

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Recall

m∗ (A) = m∗ (A ∩ E1 ) + m∗ (A ∩ E1c ∩ E2 ) + m∗ (A ∩ E1c ∩ E2c ). (10)

If F1 ∈ M, F2 ∈ M and F1 ∩ F2 = ∅ then taking

A = F 1 ∪ F 2 , E1 = F 1 , E 2 = F 2

in (10) gives
m(F1 ∪ F2 ) = m(F1 ) + m(F2 ).
Induction then shows that if F1 , . . . , Fn are pairwise disjoint
elements of M then their union belongs to M and

m(F1 ∪ F2 ∪ · · · ∪ Fn ) = m(F1 ) + m(F2 ) + · · · + m(Fn ).

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

m∗ (A) = m∗ (A ∩ E1 ) + m∗ (A ∩ E1c ∩ E2 ) + m∗ (A ∩ E1c ∩ E2c ). (10)

More generally, if we let A be arbitrary and take E1 = F1 , E2 = F2

in (10) we get
m∗ (A) = m∗ (A ∩ F1 ) + m∗ (A ∩ F2 ) + m∗ (A ∩ (F1 ∪ F2 )c ).
If F3 ∈ M is disjoint from F1 and F2 we may apply (8) with A
replaced by A ∩ (F1 ∪ F2 )c and E by F3 to get
m∗ (A ∩ (F1 ∪ F2 )c )) = m∗ (A ∩ F3 ) + m∗ (A ∩ (F1 ∪ F2 ∪ F3 )c ),
since
(F1 ∪ F2 )c ∩ F3c = F1c ∩ F2c ∩ F3c = (F1 ∪ F2 ∪ F3 )c .
Substituting this back into the preceding equation gives
m∗ (A) = m∗ (A∩F1 )+m∗ (A∩F2 )+m∗ (A∩F3 )+m∗ (A∩(F1 ∪F2 ∪F3 )c ).
Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

m∗ (A) = m∗ (A∩F1 )+m∗ (A∩F2 )+m∗ (A∩F3 )+m∗ (A∩(F1 ∪F2 ∪F3 )c ).

Proceeding inductively, we conclude that if F1 , . . . , Fn are pairwise

disjoint elements of M then
n
X
m∗ (A) = m∗ (A ∩ Fi ) + m∗ (A ∩ (F1 ∪ · · · ∪ Fn )c ). (11)
1

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Now suppose that we have a countable family {Fi } of pairwise

disjoint sets belonging to M. Since
n
!c ∞
!c
[ [
Fi ⊃ Fi
i=1 i=1

we conclude from (11) that

n ∞
!c !
X [
∗ ∗ ∗
m (A) ≥ m (A ∩ Fi ) + m A∩ Fi
1 i=1

and hence passing to the limit

∞ ∞
!c !
X [
∗ ∗ ∗
m (A) ≥ m (A ∩ Fi ) + m A∩ Fi .
1 i=1

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Now given any collection of sets Bk we can find intervals {Ik,j }

with [
Bk ⊂ Ik,j
j

and X 
m∗ (Bk ) ≤ `(Ik,j ) + .
2k
j

So [ [
Bk ⊂ Ik,j
k k,j

and hence [  X
m∗ Bk ≤ m∗ (Bk ),

the inequality being trivially true if the sum on the right is infinite.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

P∞
∩ Fk ) ≥ m∗ (A ∩ ( ∞
∗ (A
S
So i=1 m i=1 Fi )) . Thus

∞ ∞
!c !
X [
∗ ∗ ∗
m (A) ≥ m (A ∩ Fi ) + m A ∩ Fi ≥
1 i=1

∞
!! ∞
!c !
[ [
≥ m∗ A∩ Fi + m∗ A∩ Fi .
i=1 i=1

The extreme right of this inequality is the left hand side of (9)
applied to [
E= Fi ,
i

and so E ∈ M and the preceding string of inequalities must be

equalities since the middle is trapped between both sides which
must be equal.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Hence we have proved that if Fn is a disjoint countable family of

sets belonging to M then their union belongs to M and
∞
!c !
X [
m∗ (A) = m∗ (A ∩ Fi ) + m∗ A ∩ Fi . (12)
i i=1
S
If we take A = Fi we conclude that
∞
X
m(F ) = m(Fn ) (13)
n=1

if the Fj are disjoint and

[
F = Fj .

So we have reproved the last assertion of the theorem using

Caratheodory’s definition.
Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

S
Proof of 3:If En ∈ M for n = 1, 2, 3, . . . then n En ∈ M.
For the proof of this third assertion, we need only observe that a
countable union of sets in M can be always written as a countable
disjoint union of sets in M. Indeed, set

F1 := E1 , F2 := E2 \ E1 = E1 ∩ E2c

F3 := E3 \ (E1 ∪ E2 )
etc. The right hand sides all belong to M since M is closed under
taking complements and finite unions and hence intersections, and
[ [
Fj = Ej .
j

We have completed the proof of the theorem.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Some consequences - symmetric differences.

The symmetric difference between two sets is the set of points

belonging to one or the other but not both:

A∆B := (A \ B) ∪ (B \ A).

Proposition.
If A ∈ M and m(A∆B) = 0 then B ∈ M and m(A) = m(B).

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Proof.
By assumption A \ B has measure zero (and hence is measurable)
since it is contained in the set A∆B which is assumed to have
measure zero. Similarly for B \ A. Also (A ∩ B) ∈ M since

A ∩ B = A \ (A \ B).

Thus
B = (A ∩ B) ∪ (B \ A) ∈ M.
Since B \ A and A ∩ B are disjoint, we have

m(B) = m(A∩B)+m(B\A) = m(A∩B) = m(A∩B)+m(A\B) = m(A).

Shlomo Sternberg
Math212a1411 Lebesgue measure.
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More consequences: increasing limits.

Proposition.
Suppose that An ∈ M and An ⊂ An+1 for n = 1, 2, . . . . Then
[ 
m An = lim m(An ).
n→∞

S
If A := An we write the hypotheses of the proposition as An ↑ A.
In this language the proposition asserts that

An ↑ A ⇒ m(An ) → m(A).

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Proof.
Setting Bn := An \ An−1 (with B1 = A1 ) the Bi are pairwise
disjoint and have the same union as the Ai so
[  X∞ n
X
m An = m(Bi ) = lim m(Bn )
n→∞
i=1 i=1

n
!
[
= lim m Bi = lim m(An ).
n→∞ n→∞
i=1

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

More consequences: decreasing limits.

Proposition.
If Cn ⊃ Cn+1 is a decreasing family of sets in M and m(C1 ) < ∞
then \ 
m Cn = lim m(Cn ).
n→∞

Proof.
T
Let C = Cn and An := C1 /Cn . Then An is a monotone
increasing family of sets as in the preceding proposition so
m(C1 ) − m(Cn ) = m(An ) → m(C1 /C ) = m(C1 ) − m(C ). Now
subtract m(C1 ) from both sides.

Taking Cn = [n, ∞) shows that we need m(Ck ) < ∞ for some k.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
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Constantin Carathéodory

Born: 13 Sept 1873 in Berlin, Germany

Died: 2 Feb 1950 in Munich, Germany

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Math212a1411 Lebesgue measure.
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Axiomatic approach.

We will now take the items in Theorem 6 as axioms: Let X be a

set. (Usually X will be a topological space or even a metric space).
A collection F of subsets of X is called a σ field if:
X ∈ F,
If E ∈ F then E c = X \ E ∈ F, and
S
If {En } is a sequence of elements in F then n En ∈ F,
The intersection of any family of σ-fields is again a σ-field, and
hence given any collection C of subsets of X , there is a smallest
σ-field F which contains it. Then F is called the σ-field
generated by C.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Borel sets.

If X is a metric space, the σ-field generated by the collection of

open sets is called the Borel σ-field, usually denoted by B or
B(X ) and a set belonging to B is called a Borel set.

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Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Measures on a σ-field.

Given a σ-field F a (non-negative) measure is a function

m : F → [0, ∞]

such that
m(∅) = 0 and
Countable additivity: If Fn is a disjoint collection of sets in
F then !
[ X
m Fn = m(Fn ).
n n

In the countable additivity condition it is understood that both

sides might be infinite.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Outer measures.

An outer measure on a set X is a map m∗ to [0, ∞] defined on

the collection of all subsets of X which satisfies
m∗ (∅) = 0,
Monotonicity: If A ⊂ B then m∗ (A) ≤ m∗ (B), and
Countable subadditivity: m∗ ( n An ) ≤ n m∗ (An ).
S P

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Measures from outer measures via Caratheordory.

Given an outer measure, m∗ , we defined a set E to be measurable

(relative to m∗ ) if

m∗ (A) = m∗ (A ∩ E ) + m∗ (A ∩ E c )

for all sets A. Then Caratheodory’s theorem that we proved just

now asserts that the collection of measurable sets is a σ-field, and
the restriction of m∗ to the collection of measurable sets is a
measure which we shall usually denote by m.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Terminological conflict.

There is an unfortunate disagreement in terminology, in that many

of the professionals, especially in geometric measure theory, use the
term “measure” for what we have been calling “outer measure”.
However we will follow the above conventions which used to be the
old fashioned standard.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

The next task.

An obvious task, given Caratheodory’s theorem, is to look for ways

of constructing outer measures.
This will be the subject of the next lecture.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Over the next few slides I want to give remarkable applications of

our two “monotone convergence” propositions:

An ↑ A ⇒ m(An ) → m(A)

and (with the obvious notation)

Cn ↓ C and m(C1 ) < ∞ ⇒ m(Cn ) → m(C ).

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Some notation

Let En be a sequence of measurable sets. The following definitions

of the set E are synonymous:

E := {En i.o.} = {En infinitely often}

:= lim sup En
\[
:= En
k n≥k
:= {x|∀k∃n(k) ≥ k such that x ∈ En(k) }
:= {x|x ∈ En for infinitely many En }.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

The first Borel-Cantelli lemma (BC1)

Lemma
P
Suppose that m(En ) < ∞ then m(lim sup En ) = 0.

Proof.
S S
Let Gk = n≥k En . So k≤n≤p En ↑ Gk . But
 
[ X
m En  ≤ m(En )
k≤n≤p k≤n≤p
P∞
P so m(Gk ) ≤ k m(En ). Since lim sup En ⊂ Gk for all k and
and
m(En ) < ∞ we conclude that m(lim sup En ) is less than any
positive number by choosing k sufficiently large.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Some probabilistic language

We now restrict to the case where m(X ) = 1 and use P instead of

m. A measurable set is now called an event. A sequence of events
En is called independent if, for every k ∈ N, if all the i1 , . . . ik are
distinct then Y
P(Ei1 ∩ · · · ∩ Eik ) = P(Eij ).

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

The second Borel-Cantelli lemma (BC2)

Lemma
Let
P En be a sequence of independent events. Then
P(En ) = ∞ ⇒ P(En i.o.) = 1.
T S
Notice that En i.o. = k n≥k En so its complement is
[\
Enc .
k n≥k

We must show that this

T has probability zero, and for this it is
c
enough to show that n≥k En has probability zero. Now
\ \
Enc ↓ Enc
k≤n≤` n≥k
T 
so it is enough to show that P E c → 0 as ` → ∞.
k≤n≤` n
Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Proof.
Let pn := P(En ) so that P(Enc ) = 1 − pn . Then, by independence,
 
\ Y
P Enc  = (1 − pn ).
k≤n≤` k≤n≤`

Now for x ≥ 0, 1 − x ≤ e −x so
P
(1 − pn ) ≤ e − k≤n≤` pn
Y

k≤n≤`
P
and since pn = ∞ this last expression → 0.

Shlomo Sternberg
Math212a1411 Lebesgue measure.
Outline Lebesgue outer measure. Lebesgue inner measure. Lebesgue’s definition of measurability. Caratheodory’s definition of mea

Notice we get a “0 or 1 law” in that for a sequence En of

independent events we conclude that P(En infinitely often) is either
zero or one.

Shlomo Sternberg
Math212a1411 Lebesgue measure.