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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