Lecture 16 - Feb 10

Exercise: Assume Fatou’s Lemma, & prove MCT.

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Example: The inequality can be strict. Consider . Take two disjoint sets such that .

Consider for ,

Take . Then or . If then and . If , and . If then .

Then

Note: In all cases .

is odd,

is even,

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Definition (Almost Everywhere)

Suppose be measurable functions. is said to satisfy ‘property ’ almost everywhere (written as a.e.) w.r.t. if with and satisfies the ‘property ’ .

Note: In probability, it is almost surely.

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Example Let us find .

Note: Lebesgue -algebra is complete but Borel is not in any topological space.

1. is continuous almost everywhere with .

2. Define a Dirac measure

Prove that is a measure on . If is not continuous a.e. w.r.t. .

1. is continuous a.e. w.r.t. .
2. is differentiable a.e.
3. is not a.e. continuous.

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Theorem

If is a non-negative measurable function, then:

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Proof

Suppose is zero a.e. then if is a simple function & then a.e. and so .

Suppose . If possible, is not a.e. then:

Then since is not a.e. , s.t. .

Now, .

This is a contradiction since we assumed this is .

Our assumption is wrong, hence a.e.

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Infinite Sum as Lebesgue Integration

Suppose is a sequence of non-negative real numbers.

Define

And for ,

Consider the measure space where is the counting measure:

Note that

as

& .

Hence by MCT (Monotone Convergence Theorem),

Note:

if

,

Now

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