Monotone Convergence Theorem - Lebesgue's Monotone Convergence Theorem - Theorem

Theorem

Let (X, Σ, μ) be a measure space. Let be a pointwise non-decreasing sequence of -valued Σ–measurable functions, i.e. for every k ≥ 1 and every x in X,

Next, set the pointwise limit of the sequence to be f. That is, for every x in X,

Then f is Σ–measurable and

Remark. If the sequence satisfies the assumptions μ–almost everywhere, one can find a set N ∈ Σ with μ(N) = 0 such that the sequence is non-decreasing for every . The result remains true because for every k,

provided that f is Σ–measurable (see for instance section 21.38).