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I am reading Kai Lai Chung's A Course in Probability Theory. In the proof of Theorem 9.3.4 on page 341 as shown below, where $\Lambda_j=\Lambda\bigcap\{\alpha=j\}$, I do not understand how the inequality below Inequality (17) is obtained, specifically how the integrating set $\{\beta\ge j\}$ disappears or is summed away. I would appreciate it if someone can explicate it.

optional sampling theorem p.1 optional sampling theorem p.2

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    Have you taken into account that $m$ is an upper bound for $\beta$? Then in formula 17 the negative term is an integral over a set of measure zero. Then you sum over j and get and $\beta \ge j$ becomes $\beta \ge 0$ which is a set of full measure. – 2017-02-16
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    @Corn: You are right about using $m$ being an upper bound of $\beta$. $j$ is fixed, not summed over. However, $\beta\ge\alpha$. $\Lambda_j\subseteq\{\alpha=j\}\subseteq\{\beta\ge j\}$. The intersection with $\{\beta\ge j\}$ for the integral domain in Inequality (17) is thus redundant and we deduce the last inequality. – 2017-02-16

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