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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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A sufficient condition is that every $|f_n-f|$ is bounded by a single integrable function. – Did Jan 12 '12 at 10:20
Almost everywhere pointwise convergence is not enough even for $d=1$, for example taking $\Omega_n=[-(n+1),n+1]$ and $f_n=n\mathbf 1_{[n,n+1]}$. – Davide Giraudo Jan 12 '12 at 10:24
Let's say $f_n$ and $f$ are bounded ($L_\infty$). – heiner Jan 12 '12 at 10:30
Heiner: then consider $d=1$, $\Omega_n=[-n,n]$ and $f_n=\mathbf 1_{[\sqrt{n},n]}$. – Did Jan 12 '12 at 10:43
By the way, this doesn't seem very related to ergodic theory... – Mark Jan 12 '12 at 11:03

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