Convergence in measure

Convergence in measure can refer to two distinct mathematical concepts which both generalize the concept of convergence in probability.

Definitions

Let $f, f_n\ (n \in \mathbb N): X \to \mathbb R$ be measurable functions on a measure space (X,Σ,μ). The sequence (f_n) is said to converge globally in measure to f if for every ε > 0,

$\lim_{n\to\infty} \mu(\{x \in X: |f(x)-f_n(x)|\geq \varepsilon\}) = 0$,

and to converge locally in measure to f if for every ε > 0 and every $F \in \Sigma$ with $\mu (F) < \infty$,

$\lim_{n\to\infty} \mu(\{x \in F: |f(x)-f_n(x)|\geq \varepsilon\}) = 0$.

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, $\mu (X)<\infty$ or, more generally, if 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 $X = \mathbb R$, μ be Lebesgue measure, and f the constant function with value zero.

• The sequence $f_n = \chi_{[n,\infty)}$ 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 $\chi_{\left[0,1\right]},\;\chi_{\left[0,\frac12\right]},\;\chi_{\left[\frac12,1\right]},\;\chi_{\left[0,\frac14\right]},\;\chi_{\left[\frac14,\frac12\right]}$) converges to 0 locally 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 $f_n = n\chi_{\left[0,\frac1n\right]}$ converges to f almost everywhere (hence also locally in measure), but not in the p-norm for any $p \geq 1$.

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

$\{\rho_F : F \in \Sigma,\ \mu (F) < \infty\},$

where

$\rho_F(f,g) = \int_F \min\{|f-g|,1\}\, d\mu$.

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 $G\subset X$ of finite measure and $\varepsilon > 0$ there exists F in the family such that $\mu(G\setminus F)<\varepsilon.$ When $\mu(X) < \infty$, 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

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

still defines a metric that generates the global convergence in measure.^[1]

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.

References

1. Vladimir I. Bogachev, Measure Theory Vol. I, Springer Science & Business Media, 2007

• 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.