Let $X$ be random variable saistyfing $\mathbb{E}[X]=0$ and $P(X > - 1)=1$. Take $a$ be a constant such that $\log(1+aX) \geq 0$ with probability one.
If $ \mathbb{E}[\log(1+aX)]=0, $ I wonder under what condition that $a=0$.
My thought: One obvious choice is that $X=0$ with probability one, but this is not what I am looking for. I was trying to use Jensen's inequality and obtain $$ 0 = \mathbb{E}[\log(1+aX)] \leq \log(1+a\mathbb{E}[X]) = 0 $$ but this also leads me to nowhere. Any comment is appreciated.
For any random variable $Y$, if $Y\geq 0$ almost surely, then $\mathbb{E}[Y]=0$ if and only if $Y=0$ almost surely. Here, this implies $\log(1+aX)=0$ almost surely, and so $aX=0$ almost surely. This happens if and only if either $a=0$ and $X$ is an arbitrary mean zero random variable, or $a$ is arbitrary and $X=0$ almost surely. So if $X$ is not almost surely equal to $0$, you must have $a=0$.