For a positive random variable we define
$\mathbb{E}\{X\} = \sup\{\mathbb{E}\{Y\}: 0 \le Y \le X\}$, where $Y$ is a simple random variable (and we have already defined expectations for them)
Then it is written:
We can have $\mathbb{E}\{X\} = \infty$ even when $X$ is never equal to $\infty$.
Why is that so?
If $X$ is never equal to $\infty$ then $\exists M: X \le M$, and thus every simple random variable $Y$ such that $Y \le X$ is also $Y \le M \Rightarrow \mathbb{E}\{Y\} \le M$
So also $\mathbb{E}\{X\} = \sup\{\mathbb{E}\{Y\}\} \le M$
Why am I wrong?