If I know that$\ X < \sup_iX_i$ for example and if $\ X_i $ and $X$ are integrable then when I integrate both sides, where do I put the integral on the RHS? I know that in some situations you can move the supremum inside and outside according to beppo levi etc but what I want to know is forgetting that, where is it always defined or where should I always put it to start off with before thinking about beppo levi or fatou or anything else?
Integrals and suprema
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measure-theory
probability-theory
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0The supremum of a countable family of measurable functions is measurable. so you can put $\int X\leq\int\sup X_i$. Whether the integra preserves the strict inequality depends on what "<" means. If X(\omega)<\sup_i X_i(\omega) for almost all $\omega$, you can write \int X<\int\sup_i X_i, – 2012-06-07
1 Answers
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If $\displaystyle X < \sup_i X_i$ then $\displaystyle \int X \le \int \sup_i X_i$.
You can't say "$<$", but only "$\le$"; in some cases they are equal.
And there are some simple cases where $\displaystyle\sup_i \int X_i$ is not the same as $\displaystyle \int \sup_i X_i$.
For example, suppose $ X_1(x) = \begin{cases} 1 & \text{if }0\le x \le 1/2, \\ 0 & \text{if } 1/2
Then $ \sup_{i\in\{1,2\}} \int_0^1 X_i(x)\,dx = \frac 1 2 $ and $ \int \sup_{i\in\{1,2\}} X_i(x)\,dx = 1. $