Convergence in measure

Properties

Throughout, ${\displaystyle f}$ and ${\displaystyle f_n}$ (${\displaystyle n\in \mathbb{N}}$) are measurable functions ${\displaystyle X\to \mathbb{R}}$.

• 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, ${\displaystyle \mu (X)<\mathrm{\infty }}$ or, more generally, if ${\displaystyle f}$ and all the ${\displaystyle f_n}$ vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears.
• If ${\displaystyle \mu }$ is σ-finite and (f_n) converges (locally or globally) to ${\displaystyle f}$ in measure, there is a subsequence converging to ${\displaystyle f}$ almost everywhere. The assumption of σ-finiteness is not necessary in the case of global convergence in measure.
• If ${\displaystyle \mu }$ is ${\displaystyle \sigma }$-finite, ${\displaystyle (f_n)}$ converges to ${\displaystyle f}$ locally in measure if and only if every subsequence has in turn a subsequence that converges to ${\displaystyle f}$ almost everywhere.
• In particular, if ${\displaystyle (f_n)}$ converges to ${\displaystyle f}$ almost everywhere, then ${\displaystyle (f_n)}$ converges to ${\displaystyle f}$ 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 ${\displaystyle \mu }$ is ${\displaystyle \sigma }$-finite, Lebesgue's dominated convergence theorem also holds if almost everywhere convergence is replaced by (local or global) convergence in measure.
• If ${\displaystyle X=[a,b]\subseteq \mathbb{R}}$ and μ is Lebesgue measure, there are sequences ${\displaystyle (g_n)}$ of step functions and ${\displaystyle (h_n)}$ of continuous functions converging globally in measure to ${\displaystyle f}$.
• If ${\displaystyle f}$ and ${\displaystyle f_n}$ are in L^p(μ) for some ${\displaystyle p>0}$ and ${\displaystyle (f_n)}$ converges to ${\displaystyle f}$ in the ${\displaystyle p}$-norm, then ${\displaystyle (f_n)}$ converges to ${\displaystyle f}$ globally in measure. The converse is false.
• If ${\displaystyle f_n}$ converges to ${\displaystyle f}$ in measure and ${\displaystyle g_n}$ converges to ${\displaystyle g}$ in measure then ${\displaystyle f_n+g_n}$ converges to ${\displaystyle f+g}$ in measure. Additionally, if the measure space is finite, ${\displaystyle f_n{g}_n}$ also converges to ${\displaystyle fg}$.

Counterexamples

Let ${\displaystyle X=\mathbb{R}}$, ${\displaystyle \mu }$ be Lebesgue measure, and ${\displaystyle f}$ the constant function with value zero.

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

${\displaystyle f_n={\chi }_{\left[\frac{j}{2^k},\frac{j+1}{2^k}\right]},}$

where ${\displaystyle k=\lfloor {\log }_{2}n\rfloor }$ and ${\displaystyle j=n-2^k}$, the first five terms of which are

${\displaystyle {\chi }_{\left[0,1\right]},{\chi }_{\left[0,\frac{1}{2}\right]},{\chi }_{\left[\frac{1}{2},1\right]},{\chi }_{\left[0,\frac{1}{4}\right]},{\chi }_{\left[\frac{1}{4},\frac{1}{2}\right]},}$

converges to ${\displaystyle 0}$ globally in measure; but for no ${\displaystyle x}$ does ${\displaystyle f_n(x)}$ converge to zero. Hence ${\displaystyle (f_n)}$ fails to converge to ${\displaystyle f}$ almost everywhere.

• The sequence

${\displaystyle f_n=n{\chi }_{\left[0,\frac{1}{n}\right]}}$

converges to ${\displaystyle f}$ almost everywhere and globally in measure, but not in the ${\displaystyle p}$-norm for any ${\displaystyle p\ge 1}$.

Topology

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 $${\displaystyle \{{\rho }_F:F\in \mathrm{\Sigma },\mu (F)<\mathrm{\infty }\},}$$ where $${\displaystyle {\rho }_F(f,g)={\int }_{F}min\{|f-g|,1\}d\mu .}$$ 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 ${\displaystyle G\subset X}$ of finite measure and ${\displaystyle \epsilon >0}$ there exists F in the family such that ${\displaystyle \mu (G\setminus F)<\epsilon .}$ When ${\displaystyle \mu (X)<\mathrm{\infty }}$, we may consider only one metric ${\displaystyle {\rho }_X}$, so the topology of convergence in finite measure is metrizable. If ${\displaystyle \mu }$ is an arbitrary measure finite or not, then $${\displaystyle d(f,g):=\underset{\delta >0}{inf}\mu (\{|f-g|\ge \delta \})+\delta }$$ still defines a metric that generates the global convergence in measure.

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.