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Workshet 9

The document is a worksheet for Advanced Analysis at Adama Science and Technology University, containing various mathematical problems and proofs related to sequences of functions, uniform convergence, Lebesgue measure, and integrability. It includes tasks such as proving convergence of series, finding limits, and demonstrating properties of measurable functions. The problems are designed to enhance understanding of advanced mathematical concepts and their applications.

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Solomon Regasa
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52 views2 pages

Workshet 9

The document is a worksheet for Advanced Analysis at Adama Science and Technology University, containing various mathematical problems and proofs related to sequences of functions, uniform convergence, Lebesgue measure, and integrability. It includes tasks such as proving convergence of series, finding limits, and demonstrating properties of measurable functions. The problems are designed to enhance understanding of advanced mathematical concepts and their applications.

Uploaded by

Solomon Regasa
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Adama Science and Tchnology University

School of Applied Natural Science


Department of Applied Mathematics
Advanced Analysis
Worksheet II
1. Let {fn } be a sequence of continuous real valued functions defined on a closed bounded
interval [a, b] and let {an } and {bn } be two sequences in [a, b] such that lim an = a
n→∞
and lim bn = b. If {fn } converges uniformly to f on [a, b] then show that
n→∞
Z bn Z b
lim fn (x)dx = f (x)dx .
n→∞ an a

2. Prove that
a) the series
∞
X n sin(x/n)
n=1
xn + n2 ex
converges uniformly over [1, ∞).
b) the series
∞ √
X n + cos(nx)
n=1
n + n2 e x
converges uniformly over R.
c) the sequence
x −x/n
e
n
does not converge uniformly on (0, ∞).
3. If f is a non-negative continuous function on a closed bounded interval [a, b] with
f (c) > 0 for some c ∈ [a, b], then show that
Z b
f (t)dt > 0 .
a

4. If f is continuous on [a, b] prove that there exists c ∈ (a, b) such that


Z b
1
f (c) = f (t)dt .
b−a a

5. Let {fn } be a uniformly bounded sequence of continuous real valued functions defined
on a closed bounded interval [a, b]. Show that the sequence of functions defined by
Z x
ϕn (x) = fn (t)dt, for each x ∈ [a, b]
a

contains a uniformly convergent subsequence on [a, b] .


6. Find the limits and justify your calculations.
Z n
x
a) lim (1 + )−n ln(2 + cos(x/n))dx.
n→∞ 0 n
Z 1
1 + nx2
b) lim ln(2 + cos(x/n))dx.
n→∞ 0 (1 + x2 )n
Z ∞
c) lim ne−nx sin(1/x)dx.
n→∞ 0

7. Let ν be the Lebesgue measure on R . If A is Lebesgue measurable subset of R with


µ(A) > 0, then show that A has a non-measurable subset.
8. Suppose 0 <  < 1 and µ is a Lebesgue measure. Find a measurable set E ⊂ [0, 1]
such that the closure of E is [0, 1] and µ(E) = .
9. Let µ be the Lebesgue measure on R. Let E ⊂ R be Lebesgue measurable such that
0 < µ(E) < ∞. Prove that for each 0 < γ < 1, there exists an open interval I ⊂ R
such that
µ(E ∩ I) ≥ γµ(I).
10. Suppose (X, M) is a measurable space, f is a real valued function and
{x : f (x) > r} ∈ M
for each rational number r. Prove that f is measurable.
11. Let (X, M, µ) be a measure space and suppose µ is a finite measure. If f is a non-
negative integrable function on X, prove that given  > 0 there exists δ > 0 such
that Z
f dµ < 
A
whenever µ(A) < δ.
12. If fn is a sequence of non-negative
R integrable
R functions such that fn (x) decreases to
f (x) for every x. Prove that fn dµ → f dµ.
13. Let g : R → R be integrable and let f : R → R be bounded measurable and continuous
at 1. Prove that Z n
lim f (1 + x/n2 )g(x)dx
n→∞ −n
exists and find its value.
14. Suppose (X, M, µ) is Ra measure
R space, each fn is integrable non-negative function such
that fn → f a.e and fn → f. Prove that for each A ∈ M,
Z Z
fn → f.
A A
R R R
15. If fn , f are integrable fn → f a.e and |fn | → |f |, then show that |fn − f | → 0.

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