Convergence in measure

Let $f,\; f\_n\backslash \; (n\; \backslash in\; \backslash mathbb\; N):\; X\; \backslash to\; \backslash mathbb\; R$ be measurable functions on a measure space $(X,\; \backslash Sigma,\; \backslash mu).$ The sequence $f\_n$ is said to {{visible anchor|converge globally in measure|Global convergence in measure}} to $f$ if for every $\backslash varepsilon\; >\; 0,$

$$\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash mu(\backslash \{x\; \backslash in\; X:\; |f(x)-f\_n(x)|\backslash geq\; \backslash varepsilon\backslash \})\; =\; 0,$$

and to {{visible anchor|converge locally in measure|Local convergence in measure}} to $f$ if for every $\backslash varepsilon>0$ and every $F\; \backslash in\; \backslash Sigma$ with

$$\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash mu(\backslash \{x\; \backslash in\; F:\; |f(x)-f\_n(x)|\backslash geq\; \backslash varepsilon\backslash \})\; =\; 0.$$

On a finite measure space, both notions are equivalent. Otherwise, convergence in measure can refer to either global convergence in measure{{cite book |last1=Bogachev |first1=Vladimir Igorevich |title=Measure theory |date=2007 |publisher=Springer |location=Berlin New York |isbn=978-3-540-34514-5}}{{rp|2.2.3}} or local convergence in measure, depending on the author.

Throughout, $f$ and $f\_n$ ($n\backslash in\backslash N$) are measurable functions $X\backslash to\backslash R$.

• Global convergence in measure implies local convergence in measure. The converse, however, is false; i.e., local convergence in measure is strictly weaker than global convergence in measure, in general.
• If, however, or, more generally, if $f$ and all the $f\_n$ vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears.
• If $\backslash mu$ is σ-finite and (f_n) converges (locally or globally) to $f$ in measure, there is a subsequence converging to $f$ almost everywhere.{{rp|2.2.5}} The assumption of σ-finiteness is not necessary in the case of global convergence in measure.
• If $\backslash mu$ is $\backslash sigma$-finite, $(f\_n)$ converges to $f$ locally in measure if and only if every subsequence has in turn a subsequence that converges to $f$ almost everywhere.
• In particular, if $(f\_n)$ converges to $f$ almost everywhere, then $(f\_n)$ converges to $f$ locally in measure. The converse is false.
• Fatou's lemma and the monotone convergence theorem hold if almost everywhere convergence is replaced by (local or global) convergence in measure.
• If $\backslash mu$ is $\backslash sigma$-finite, Lebesgue's dominated convergence theorem also holds if almost everywhere convergence is replaced by (local or global) convergence in measure.{{rp|2.8.6}}
• If $X=[a,b]\backslash subseteq\backslash R$ and μ is Lebesgue measure, there are sequences $(g\_n)$ of step functions and $(h\_n)$ of continuous functions converging globally in measure to $f$.
• If $f$ and $f\_n$ are in L^p(μ) for some $p>0$ and $(f\_n)$ converges to $f$ in the $p$-norm, then $(f\_n)$ converges to $f$ globally in measure. The converse is false.
• If $f\_n$ converges to $f$ in measure and $g\_n$ converges to $g$ in measure then $f\_n+g\_n$ converges to $f+g$ in measure. Additionally, if the measure space is finite, $f\_n\; g\_n$ also converges to $fg$.

Let $X\; =\; \backslash Reals$, $\backslash mu$ be Lebesgue measure, and $f$ the constant function with value zero.

• The sequence $f\_n\; =\; \backslash chi\_\{[n,\backslash infty)\}$ converges to $f$ locally in measure, but does not converge to $f$ globally in measure.
• The sequence

::$f\_n\; =\; \backslash chi\_\{\backslash left[\backslash frac\{j\}\{2^k\},\backslash frac\{j+1\}\{2^k\}\backslash right]\},$

:where $k\; =\; \backslash lfloor\; \backslash log\_2\; n\backslash rfloor$ and $j=n-2^k$, the first five terms of which are

::$\backslash chi\_\{\backslash left[0,1\backslash right]\},\; \backslash ;\backslash chi\_\{\backslash left[0,\backslash frac12\backslash right]\},\backslash ;\backslash chi\_\{\backslash left[\backslash frac12,1\backslash right]\},\backslash ;\backslash chi\_\{\backslash left[0,\backslash frac14\backslash right]\},\backslash ;\backslash chi\_\{\backslash left[\backslash frac14,\backslash frac12\backslash right]\},$

:converges to $0$ globally in measure; but for no $x$ does $f\_n(x)$ converge to zero. Hence $(f\_n)$ fails to converge to $f$ almost everywhere.{{rp|2.2.4}}

• The sequence

::$f\_n\; =\; n\backslash chi\_\{\backslash left[0,\backslash frac1n\backslash right]\}$

:converges to $f$ almost everywhere and globally in measure, but not in the $p$-norm for any $p\; \backslash geq\; 1$.

There is a topology, called the topology of (local) convergence in measure, on the collection of measurable functions from X such that local convergence in measure corresponds to convergence on that topology.

This topology is defined by the family of pseudometrics

where

$$\backslash rho\_F(f,g)\; =\; \backslash int\_F\; \backslash min\backslash \{|f-g|,1\backslash \}\backslash ,\; d\backslash mu.$$

In general, one may restrict oneself to some subfamily of sets F (instead of all possible subsets of finite measure). It suffices that for each $G\backslash subset\; X$ of finite measure and $\backslash varepsilon\; >\; 0$ there exists F in the family such that When , we may consider only one metric $\backslash rho\_X$, so the topology of convergence in finite measure is metrizable. If $\backslash mu$ is an arbitrary measure finite or not, then

$$d(f,g)\; :=\; \backslash inf\backslash limits\_\{\backslash delta>0\}\; \backslash mu(\backslash \{|f-g|\backslash geq\backslash delta\backslash \})\; +\; \backslash delta$$

still defines a metric that generates the global convergence in measure.Vladimir I. Bogachev, Measure Theory Vol. I, Springer Science & Business Media, 2007

Because this topology is generated by a family of pseudometrics, it is uniformizable.

Working with uniform structures instead of topologies allows us to formulate uniform properties such as

Cauchyness.

• Convergence space

{{reflist}}

• D.H. Fremlin, 2000. [https://web.archive.org/web/20101101220236/http://www.essex.ac.uk/maths/people/fremlin/mt.htm Measure Theory]. Torres Fremlin.
• H.L. Royden, 1988. Real Analysis. Prentice Hall.
• G. B. Folland 1999, Section 2.4. Real Analysis. John Wiley & Sons.

{{Measure theory}}

{{Lp spaces}}

Category:Measure theory

Measure, Convergence in

Category:Lp spaces