Lecture 12 - Jan 30

Remark:

1. Any simple function is measurable.

2. is simple range of is finite & is measurable.

If range then we take & .

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Definition

Let be a non-negative measure space & is a non-negative simple fn with representation .

Then the integral of wrt is defined as

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Well definedness: Suppose . Then .

Need to prove that .

If then . We write

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Proposition

Let and be two measurable functions on . Then

i)

ii)

iii) If then

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Proof of ii): ,

We write,

Note: where

Similarly,

Note: ,

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iii)

Since are simple then is simple.

With standard representation:

Using (iii), we can write .

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where is a non-negative simple fn.

is a non-negative measurable fn,

Note: , ,

of simple fn

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