Basic question about the probability and expectation of a bijective function.

Question

Dear stackexchange community,

I am still unskilled in the language of mathematics, in fact probability theory to be precise. In my spare time I like to do some research of my own, and I am having difficulties proving the following: $ P(f(X)>a)=P(X>f^{-1}(a))\\ E[f(X)|f(X)<a]=E[f(X)|X<f^{-1}(a)]$

The attempt I made for the first proof is: $X \ is \ R.V. \ f \ strictly \ increasing \ \\ P(f(X)>a)=P(X>f^{-1}(a))\\ \Rightarrow P(f(X)>a)=E[I_{[a,\infty]}f(X)]= E[I_{[f^{-1}(a)),f^{-1}(\infty)]}f^{-1}f(X)]\\ =E[I_{[f^{-1}(a)),\infty]}X]=P(X>f^{-1}(a))$

but I highly doubt the above.

Moreover, I cannot figure out why $E[f(X)|f(X)<a]=E[f(X)|X<f^{-1}(a)]$ holds true.

Can you guys help me out, and perhaps recommend some literature that deepens my knowledge of the subject?

Thanks in advance!

Answer 1

All your questions are solved by the remark that, if $f$ is bijective and increasing, then $A=B$, where $A=[f(X)\lt a]$ and $B=[X\lt f^{-1}(a)]$.

Hence $P(A)=P(B)$ and $E(f(X)\mid A)=E(f(X)\mid B)$.