Convergence in measure

Convergence in measure is either of two distinct mathematical concepts both of which generalize the concept of convergence in probability.

Definitions

Let $f,f_n(n\in \mathbb{N}):X\to ℝ$ be measurable functions on a measure space $(X,\mathrm{\Sigma },\mu ).$ The sequence $f_n$ is said to Template:Visible anchor to $f$ if for every $\epsilon >0,$ $$\underset{n\to \mathrm{\infty }}{\lim }\mu (\{x\in X:|f(x)-f_n(x)|\ge \epsilon \})=0,$$ and to Template:Visible anchor to $f$ if for every $\epsilon >0$ and every $F\in \mathrm{\Sigma }$ with $\mu (F)<\mathrm{\infty },$ $$\underset{n\to \mathrm{\infty }}{\lim }\mu (\{x\in F:|f(x)-f_n(x)|\ge \epsilon \})=0.$$

On a finite measure space, both notions are equivalent. Otherwise, convergence in measure can refer to either global convergence in measure or local convergence in measure, depending on the author.

Properties

Throughout, f and f_n (n $\in$ N) are measurable functions X → 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, $\mu (X)<\mathrm{\infty }$ 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 μ is σ-finite and (f_n) converges (locally or globally) to f in measure, there is a subsequence converging to f almost everywhere. The assumption of σ-finiteness is not necessary in the case of global convergence in measure.
• If μ is σ-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 μ is σ-finite, Lebesgue's dominated convergence theorem also holds if almost everywhere convergence is replaced by (local or global) convergence in measure.
• If X = [a,b] ⊆ 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 (n ∈ 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_ng_n also converges to fg.

Counterexamples

Let $X=ℝ$ μ be Lebesgue measure, and f the constant function with value zero.

• The sequence $f_n={\chi }_{[n,\mathrm{\infty })}$ converges to f locally in measure, but does not converge to f globally in measure.
• The sequence $f_n={\chi }_{[\frac{j}{2^k},\frac{j+1}{2^k}]}$ where $k=\lfloor {\log }_{2}n\rfloor$ and $j=n-2^k$ (The first five terms of which are ${\chi }_{[0,1]},{\chi }_{[0,\frac{1}{2}]},{\chi }_{[\frac{1}{2},1]},{\chi }_{[0,\frac{1}{4}]},{\chi }_{[\frac{1}{4},\frac{1}{2}]}$) 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.

• The sequence $f_n=n{\chi }_{[0,\frac{1}{n}]}$ converges to f almost everywhere and globally in measure, but not in the p-norm for any $p\ge 1$.

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

See also

• Convergence space

References

• D.H. Fremlin, 2000. Measure Theory. Torres Fremlin.
• H.L. Royden, 1988. Real Analysis. Prentice Hall.
• G. B. Folland 1999, Section 2.4. Real Analysis. John Wiley & Sons.