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Lebesgue Integration

The document discusses Lebesgue integration and the Lebesgue integral. It defines what it means for a function to be Lebesgue integrable over a set of finite measure. Specifically, a bounded and measurable function is Lebesgue integrable if its upper and lower integrals are equal. Theorems are presented about linearity, monotonicity, and additivity of integration. Boundedness, measurability, and finite measure are required for a function to have a Lebesgue integral.

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Engdasew Birhane
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198 views17 pages

Lebesgue Integration

The document discusses Lebesgue integration and the Lebesgue integral. It defines what it means for a function to be Lebesgue integrable over a set of finite measure. Specifically, a bounded and measurable function is Lebesgue integrable if its upper and lower integrals are equal. Theorems are presented about linearity, monotonicity, and additivity of integration. Boundedness, measurability, and finite measure are required for a function to have a Lebesgue integral.

Uploaded by

Engdasew Birhane
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Lebesgue Integration

THE RIEMANN INTEGRAL


• Define the lower and upper Darboux stuns for f
with respect to P, respectively, by
THE RIEMANN INTEGRAL
• if the two are equal we say that f is Riemann integrable over [a, b] and
call this common value the Riemann integral of f over [a, b]. We denote
it by
The Lebesgue Integral of A Bounded Measurable Function Over A Set of
Finite Measure

• The canonical representation of𝜑 on E as


The Lebesgue Integral of A Bounded Measurable Function Over A Set of
Finite Measure

• Definition A bounded function f on a domain E of finite measure is said


to be Lebesgue integrable over E provided its upper and lower Lebesgue
integrals over E are equal. The common value of the upper and lower
integrals is called the Lebesgue integral, or simply the integral, off over E

and is denoted by 𝐸
𝑓.
The Lebesgue Integral of A Bounded Measurable Function Over A Set of
Finite Measure

• Theorem : Let f be a bounded measurable function on a set


of finite measure E. Then f is integrable over E.
• Theorem : (Linearity and Monotonicity of Integration) Let f
and g be bounded measurable functions on a set of finite
measure E. Then for any 𝛼 and 𝛽,
The Lebesgue Integral of A Bounded Measurable Function Over A Set of
Finite Measure

• Corollary: Let f be a bounded measurable function on a set of


finite measure E. Suppose A and B are disjoint measurable subsets
of E. Then
The Lebesgue Integral of A Bounded Measurable Function Over A Set of
Finite Measure

• Corollary : Let f be a bounded measurable function on a set


of finite measure E. Then
The Lebesgue Integral of A Bounded Measurable Function Over A Set of
Finite Measure

•
The Lebesgue Integral of A Bounded Measurable Function Over A Set of
Finite Measure

• Therefore, by the above Corollary and the monotonicity of


integration
The Lebesgue Integral of A Bounded Measurable Function Over A Set of
Finite Measure
• Fatou's Lemma Let {𝑓𝑛 } be a sequence of nonnegative measurable
functions on E.
The General Lebesgue Integral

• For an extended real-valued function f on E, we have defined


the positive part 𝑓 + and the negative part 𝑓 − of f,
respectively, by
The General Lebesgue Integral

• Definition A measurable function f on E is said to be integrable over E


provided |𝑓| is integrable over E. When this is so we define the integral
off over E by
The General Lebesgue Integral

•
The General Lebesgue Integral

• Fatou's Lemma tells us that


The General Lebesgue Integral

• Theorem (General Lebesgue Dominated Convergence


Theorem) Let { fn} be a sequence of measurable functions
on E that converges pointwise a.e. on E to f. Suppose there is
a sequence {gn } of nonnegative measurable functions on E
that converges pointwise a.e. on E to g and dominates {𝑓𝑛 }
on E in the sense that
The General Lebesgue Integral

• Theorem (the Continuity of Integration) Let f be


integrable over E.

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