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Let $(X, \Sigma, \mu)$ a measurable space and $f$ an integrable function. Show that if $(F_n)_{n\in\mathbb N}$ is a decreasing sequence of measurable sets and $F=\bigcap_{n} F_n$, then

$$\int_{F}fd\mu = \lim_{n \to \infty} \int_{F_n}fd\mu$$

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    Hint: assume $f \geq 0$ and consider the finite measure $d\nu = f d\mu$.2012-09-16
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    Hi LFRC, welcome to Math.SE. This looks like a homework question, and for such questions we ask that you follow certain guidelines; see the FAQ at http://meta.math.stackexchange.com/questions/1803/how-to-ask-a-homework-question. In particular, simply stating a problem without any indication of what you tried, what you know, or which part is causing you trouble, is not acceptable.2012-09-16
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    I'm so sorry, but I'm new in the website, I will follow the guidelines since now. What I've tried is to redefine the sequence in this way: $E_1 = F_1, E_2 = F_2 - F_1, ...E_n =F_n - F_{n-1}$, later I tried to make the integral over the characteristic function for use the convergence dominated theorem but I'm a few confused about this way, I'm ok?2012-09-16

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