Interesting Career Personality Test (Free): https://mathtuition88.com/free-career-quiz/
Let be a measure space, and let be a measurable function. Define the map , , where denotes the characteristic function of .
(a) Show that is a measure and that it is absolutely continuous with respect to .
(b) Show that for any measurable function , one has in .
Proof: For part (a), we routinely check that is indeed a measure.
. Let be mutually disjoiint measurable sets.
If , then a.e., thus . Therefore .
(b) We note that when is a characteristic function, i.e. ,
Hence the equation holds. By linearity, we can see that the equation holds for all simple functions. Let be a sequence of simple functions such that . Then by the Monotone Convergence Theorem, .
Note that , thus by MCT, . Note that . Hence, , and we are done.