====== Lecture 7 ======
[[https://berkeley.zoom.us/rec/share/SdfWnClv3gWmPgil0YBGjz6R8VjIzOxgjXxBtDVBYihAlqEmNgO05vYOfJs_B6xg.6sYZJmavSHrJtUXv | video]]

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