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Lecture 26 - Mar 23

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Let be a measure space.

Definition (Essentially Bounded)

A measurable function is called essentially bounded if there exists such that

If , then

Definition ( and Essential Supremum Norm)

Define

Example / Facts

is a complex vector space and is a norm on .

Facts

  1. The infimum in the definition of is attained:
  1. If , then
  1. Endpoint Holder case (check): if we interpret (so , ), then for and ,

Theorem

is complete with respect to .

Proof

Suppose is a Cauchy sequence in . We need to show there exists such that

For each , there exists such that

Hence for each , there is with such that for all ,

Define

Then , and for each ,

Thus is Cauchy in for each .

Define

Then is measurable.

Now fix and keep fixed; taking in , we get

So

hence .

Also,

for large , so .

Therefore is complete.

Remarks

  1. If is any function, then for all ,
  1. If and are sequences of non-negative simple functions such that and pointwise, then
  1. If , then there exist sequences of non-negative simple functions such that

Define

Then pointwise and .

Theorem (Density of Finite-Measure Simple Functions)

Let . Define

Then is a dense subspace of .

If , say , then

Proof

It is clear that is a subspace of .

Given , choose such that

Also,

Since a.e., by DCT,

that is,

Hence is dense in .

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