Convergence in Measure - Properties

Properties

Throughout, f and f_n (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, 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 = ⊆ 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 Lp(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.