math105-s22:notes:lecture_7

We will first follow Pugh's approach, then we will cover Tao's approach in exercises.

- Use undergraph of a non-negative function to define measurability and its measure. If the measure is finite, then call this function integrable.
- Monotone convergence theorem. (Recall upward/downward continuity theorem)
- Completed undergraph (not the closure of the undergraph, but just fiberwise closure). Can be used interchangeably with the undergraph
- upper and lower envelope sequence of a function, just like how one define the liminf and limsup.
- Dominated Convergence theorem.
- Many examples: running bump, shrinking bumps.

Discussion question:

- In Tao, one define measurable function $f: \R \to \R$, to be such that pre-image of open sets are measurable. Does this agree with Pugh's definition using undergraph?
- Pugh Ex 25, 28

math105-s22/notes/lecture_7.txt · Last modified: 2022/02/08 21:41 by pzhou