probability theory - Monotone Convergence Theorem - Lebesgue measure

One of the conditions for the monotone convergence theorem is that $f_n \uparrow f$ pointwise. Is there a version of this theorem for which $f_n \downarrow f$ pointwise? If there is, what are the other conditions?

Any help would be appreciated.

Answer

Yes. Look up dominated convergence. Basically, when approaching from above, you need for the sequence of functions to eventually have finite integral, then you can do a subtraction to get out monotone convergence. If the sequence always has infinite integral, it could converge to anything, imagine $f_n=1_{[n,\infty]}$, for example.