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Homemodes of convergence of sequences of measurable functions
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modes of convergence of sequences of measurable functions
Let $(X,\mathfrak{B},\mu)$ be a measure space, $f_{n}\colon X\to[\infty,\infty]$ be measurable functions for every positive integer $n$, and $f\colon X\to[\infty,\infty]$ be a measurable function. The following are modes of convergence of $\{f_{n}\}$:

$\{f_{n}\}$ converges almost everywhere to $f$ if $\displaystyle\mu\left(X\{x\in X:\lim_{{n\to\infty}}f_{n}(x)=f(x)\}\right)=0$

$\{f_{n}\}$ converges almost uniformly to $f$ if, for every $\varepsilon>0$, there exists $E_{\varepsilon}\in\mathfrak{B}$ with $\mu(XE_{\varepsilon})<\varepsilon$ and $\{f_{n}\}$ converges uniformly to $f$ on $E_{\varepsilon}$

$\{f_{n}\}$ converges in measure to $f$ if, for every $\varepsilon>0$, there exists a positive integer $N$ such that, for every positive integer $n\geq N$, $\displaystyle\mu\left(\{x\in X:f_{n}(x)f(x)\geq\varepsilon\}\right)<\varepsilon$.

If, in addition, $f$ and each $f_{n}$ are also Lebesgue integrable, $\{f_{n}\}$ converges in $L^{1}(\mu)$ to $f$ if $\displaystyle\lim_{{n\to\infty}}\int_{X}\leftf_{n}f\right\,d\mu=0$.
A lot of theorems in real analysis deal with these modes of convergence. For example, Fatou’s lemma, Lebesgue’s monotone convergence theorem, and Lebesgue’s dominated convergence theorem give conditions on sequences of measurable functions that converge almost everywhere under which they also converge in $L^{1}(\mu)$. Also, Egorov’s theorem states that, if $\mu(X)<\infty$, then convergence almost everywhere implies almost uniform convergence.
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