Convergence in measure is either of two distinct mathematical concepts both of which generalize the concept of convergence in probability.
Let f , f n ( n ∈ N ) : X → R be measurable functions on a measure space (X,Σ,μ). The sequence (fn) is said to converge globally in measure to f if for every ε > 0,
lim n → ∞ μ ( { x ∈ X : | f ( x ) − f n ( x ) | ≥ ε } ) = 0 ,
and to converge locally in measure to f if for every ε > 0 and every F ∈ Σ with μ ( F ) < ∞ ,
lim n → ∞ μ ( { x ∈ F : | f ( x ) − f n ( x ) | ≥ ε } ) = 0 .
Convergence in measure can refer to either global convergence in measure or local convergence in measure, depending on the author.
Throughout, f and fn (n ∈ 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, μ ( X ) < ∞ or, more generally, if all the fn vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears.If μ is σ-finite and (fn) 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, (fn) 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 (fn) converges to f almost everywhere, then (fn) 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 (gn) of step functions and (hn) of continuous functions converging globally in measure to f. If f and fn (n ∈ N) are in Lp(μ) for some p > 0 and (fn) converges to f in the p-norm, then (fn) converges to f globally in measure. The converse is false.If fn converges to f in measure and gn converges to g in measure then fn + gn converges to f + g in measure. Additionally, if the measure space is finite, fngn also converges to fg.Let X = R , μ be Lebesgue measure, and f the constant function with value zero.
The sequence f n = χ [ n , ∞ ) converges to f locally in measure, but does not converge to f globally in measure.The sequence f n = χ [ j 2 k , j + 1 2 k ] where k = ⌊ log 2 n ⌋ and j = n − 2 k (The first five terms of which are χ [ 0 , 1 ] , χ [ 0 , 1 2 ] , χ [ 1 2 , 1 ] , χ [ 0 , 1 4 ] , χ [ 1 4 , 1 2 ] ) converges to 0 globally in measure; but for no x does fn(x) converge to zero. Hence (fn) fails to converge to f almost everywhere.
The sequence f n = n χ [ 0 , 1 n ] converges to f almost everywhere and globally in measure, but not in the p-norm for any p ≥ 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
{ ρ F : F ∈ Σ , μ ( F ) < ∞ } , where
ρ F ( f , g ) = ∫ F min { | f − g | , 1 } d μ .
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 ⊂ X of finite measure and ε > 0 there exists F in the family such that μ ( G ∖ F ) < ε . When μ ( X ) < ∞ , we may consider only one metric ρ X , so the topology of convergence in finite measure is metrizable. If μ is an arbitrary measure finite or not, then
d ( f , g ) := inf δ > 0 μ ( { | f − g | ≥ δ } ) + δ 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.
