Lecture 14 - Feb 4

By def:

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Note: By def :- , if is simple.

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Textbook: Measure Theory & Integration, G. de Barra, New Age Publication

Definitions

If is measurable and we define

Proposition

If , then for any measurable

Proof:

It is enough to show for all non-negative simple functions.

Take

is a simple fn.

Proposition

Let be a non-negative simple fn. Then, [is a measure].

Then is a measure on the -field .

Proof: (trivial)

Need to prove if is a seq. of disjoint sets in .

Then

Suppose

(Complete the proof)

Monotone Convergence Theorem

Let be a sequence of non-negative measurable functions and let . Then

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Take to be a enumeration of rationals in

Note: is Borel -algebra.

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Proposition

If and are two non-negative measurable functions then

2. If then

Proof: ii)

(Try to prove i)

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