If a random variable is bounded by a constant with high probability, is its expectation also bounded by the same constant with the same probability? Suppose $X$ is a random variable that is bounded with high probability, i.e., $|X| < M$ for some $M \in \mathbb{R}^+$ with probability $1-p$. Is it correct to say that $\mathbb{E}(|X|)<M$ with probability $1 - p$ as well?

Let $X$ be a random variable such that:
$$
P(X=0)=0.999\\
P(X=1000000)=0.001
$$
Also, let $M=1$.
Then $P(\vert X\vert <M)=0.999$, which I say counts as a high probability. However, $\mathbb E[X]=1000>1$.
