Definition

Let be a measurable space. For a non-negative measurable function define

Remark: If takes values then we denote

Theorem

Let be a measurable space. be a measurable function. Then a sequence of non-negative simple functions such that

Proof: i)

Fix

Define

Observe that

Fix , If then . Then . Therefore as

If . Then there exist

we fix , Then st

iii) If is bounded then for all for some

Then

ii) Fix . We need to prove [is non-decreasing]

Case I: Assume then and .

Case II: Assume . Then

Case B: Assume . s.t.

If ,