Measure, Integration & Real Analysis · Problem 6B.5

Banach Spaces — Sheldon Axler

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

Problem

Suppose

(X, \mathcal{S})

is a measurable space and

f : X \to \mathbf{C}

is a complex-valued function. For conditions (b) and (c) below, identify

\mathbf{C}

with

\mathbf{R}^2

. Prove that the following are equivalent. (a)

f

\mathcal{S}

-measurable. (b)

f^{-1}(G) \in \mathcal{S}

for every open set

G

\mathbf{R}^2

. (c)

f^{-1}(B) \in \mathcal{S}

for every Borel set

B \in \mathcal{B}_2