Is there a conditional Markov inequality? I mean, assume that $X$ is a random variable on the probability space $(\Omega, \mathcal{A}, \mathbb{P})$, and $X\geqslant 0$. Is then
$\mathbb{P}(X\geqslant a \mid \mathcal{F})\leqslant \frac{1}{a}\mathbb{E}(X\mid \mathcal{F}),$
with $\mathcal{F}\subseteq \mathcal{A}$ and $a> 0$?
I tried to prove that by looking at the inequality $a\mathbb{1}_{X\geqslant a}\mathbb{1}_A\leqslant X\mathbb{1}_A$ for all $A\in \mathcal{F}$. If I use the expectation of this inequality, the desired result follows. Is this correct?