# 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 $\mu(X-E_{\varepsilon})<\varepsilon$ 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 , $f$ and each $f_{n}$ are also Lebesgue integrable  , $\{f_{n}\}$ converges in $L^{1}(\mu)$ to $f$ if $\displaystyle\lim_{n\to\infty}\int_{X}\left|f_{n}-f\right|\,d\mu=0$.

A lot of theorems in real analysis (http://planetmath.org/BibliographyForRealAnalysis) 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 that, if $\mu(X)<\infty$, then convergence almost everywhere implies almost uniform convergence  .

Title modes of convergence of sequences of measurable functions ModesOfConvergenceOfSequencesOfMeasurableFunctions 2013-03-22 16:14:05 2013-03-22 16:14:05 Wkbj79 (1863) Wkbj79 (1863) 7 Wkbj79 (1863) Definition msc 28A20 TravelingHumpSequence VitaliConvergenceTheorem converges almost everywhere convergence almost everywhere converges almost uniformly almost uniform convergence converges in measure convergence in measure converges in $L^{1}(\mu)$ $L^{1}(\mu)$ convergence