Suppose that $X$ is a random variable such that $\operatorname EX=0$ and $\operatorname E|X|^s<\infty$ with $s>1$. Suppose that $A$ is an event with $p=P(A)>0$ and consider a random variable $$ X'=XI_{A^c}+p^{-1}\operatorname E[XI_A]I_A, $$ where $A^c$ is the complement of $A$ and $I_A$ is the indicator of the event $A$.
Is it true that $\operatorname E|X'|^s\le\operatorname E|X|^s$?
By Hölder's inequality, $\operatorname E|XI_A|\le(\operatorname E|X|^s)^{1/s}p^{1-1/s}$ for $s>1$. It follows that \begin{align*} \operatorname E|X'|^s &=\int_{A^c}|X(\omega)|^sdP(\omega)+\int_A|p^{-1}\operatorname E[XI_A]|^sdP(\omega)\\ &=\operatorname E[|X|^sI_{A^c}]+p^{1-s}|\operatorname E[XI_A]|^s\\ &\le\operatorname E[|X|^sI_{A^c}]+\operatorname E|X|^s\\ &\le2\operatorname E|X|^s. \end{align*} Hence, I can show that $\operatorname E|X'|^s\le2\operatorname E|X|^s$. But can I drop that $2$ in front of the expected value?
Any help is much appreciated!