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Let $(\Omega_n)$ be a sequence of subsets of $\mathbb{R}^d$ with $\Omega_n\uparrow\mathbb{R}^d$ where the Lebesgue measure $\lambda^d(\Omega_n)$ is finite for every $n$. Let $(f_n)$ be a sequence of measurable functions on $\mathbb{R}^d$ with $f_n\to f$ in some sense, for instance pointwise almost everywhere. Under which conditions is it true that $\lim_{n\to\infty}\frac{1}{\lambda^d(\Omega_n)}\int_{\Omega_n} f_n(x) dx = \lim_{n\to\infty}\frac{1}{\lambda^d(\Omega_n)}\int_{\Omega_n} f(x)dx$

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    By the way, this doesn't seem very related to ergodic theory...2012-01-12

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