Suppose that $X$ is an unbounded random variable such that $\operatorname EX=0$ and $\operatorname E|X|^q=1$ with some $q>2$. Only the distribution of $X$ matters, so the probability space can be chosen freely. Choose any $p\in(0,1)$. I want to prove that there exists an event $F$ such that

- $P(F)=p$;
- $|X'|\le p^{-1/q}$, where $X'=X\chi_{F^c}+p^{-1}\operatorname E[X\chi_F]\chi_F$.
How can I prove that such an event exists?

Since $|X'|$ has to be $\le p^{-1/q}$, this means that $|X(\omega)|\le p^{-1/q}$ when $\omega\in F^c$. Using Jensen's inequality and the assumption that $\operatorname E|X|^q=1$, $|p^{-1}\operatorname E[X\chi_F]|\le p^{-1/q}$.

As a starting point, consider $G=\{|X|>p^{-1/q}\}$. Clearly, $|X(\omega)|\le p^{-1/q}$ when $\omega\in G^c$. However, using Markov's inequality and the fact that $\operatorname E|X|^q=1$, we only have that $P(G)\le p$. So $G$ only satisfies one of the required conditions.

Consider $\tilde G$ which is obtained by taking some $\omega$'s from $G^c$ and putting them into $G$ so that $G\subset\tilde G$. Since $\tilde G^c\subset G^c$, we have that $|X(\omega)|\le p^{-1/q}$ when $\omega\in\tilde G^c$. Since $G\subset\tilde G$, $P(\tilde G)\ge P(G)$. So perhaps if we take, loosely speaking, the right amount of $\omega$'s from $G^c$ and put them into $G$, we might have that $P(\tilde G)=p$ and $|X(\omega)|\le p^{-1/q}$ when $\omega\in\tilde G^c$. Is it possible to make this argument rigorous?

Maybe there is a better and simpler way to prove the existence of such an event. Any help is much appreciated.

P.S. This is a part of a very nice answer that I have received previously and I am trying to understand every part of it.