Let $e_{x} = \int_{0}^{\infty} p_{x}(t) \ dt$ where $p_{x}(t)$ is the probability that a person aged $x$ will survive at least $t$ more years. Why is $e_{x} \leq e_{x+1}+1$?
We know that $e_{x} \geq e_{x+1}$ since the expected future lifetime of a younger person is greater than that of an older person. So maybe we can show that $e_{x}-e_{x+1} \leq 1$ or $\int_{0}^{\infty} p_{x}(t) - p_{x+1}(t) \ dt \leq 1$
Edit. We know that for a discrete random variable, $\mathbb{E}(Y|X) = \int_{\Omega} Y(\omega) \mathbb{P}(dw|X=x)$
$= \frac{\int_{X=x} Y(\omega)\mathbb{P}(dw)}{\mathbb{P}(X=x)}$
$= \frac{\mathbb{E}(Y \mathbb{1}_{(X=x)})}{\mathbb{P}(X=x)}$
What does $w$ above represent?