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Hi: I need assistance with the following problem:

Let $(X,M,\mu)$ be a measure space. Let $X$ be the union of a countable ascending sequence of measurable sets $\{X_n\}$ and $f$ a nonnegative measurable function on $X$. Show that $f$ is integrable over $X$ if and only if there is an $N \geqslant 0$ for which $\int_{X_n} f~\text{d}\mu \leqslant N$ for all $n$.

I think I have to use these facts: Since $X_n\subset X_{n+1}$ and $X=\cup X_n$, $\mu(X) = \lim X_n$. For one direction, I know I have to show $\int_X fd\mu <\infty$.

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    Try proving the contrapositive. It should pop right out.2012-03-08
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    ($\Rightarrow$) is obvious, just use $N = \int_X f\ d\mu$. ($\Leftarrow$) is less obvious. Set $f_n = f$ on $X_n$ and $0$ elsewhere. Because $X_n$ are measurable, $f_n$ is also measurable and nonnegative, and then you could just use Fatou's lemma.2012-03-08
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    @dtldarek: how do I use the condition that $X=\cup X_n$?2012-03-08
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    @Jacob You need $\lim_n f_n = f$, and for that both $X_n \subset X_{n+1}$ and $X = \bigcup_n X_n$.2012-03-08

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