Measure, Integration & Real Analysis · Problem 6B.7

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · measurability-limit-complex-valued

Problem

Suppose

(X, \mathcal{S})

is a measurable space and

f_1, f_2, \ldots

is a sequence of

\mathcal{S}

-measurable functions from

X

\mathbf{C}

. Suppose

\lim_{k \to \infty} f_k(x)

exists for each

x \in X

. Define

f : X \to \mathbf{C}

f(x) = \lim_{k \to \infty} f_k(x).

Prove that

f

is an

\mathcal{S}

-measurable function.