Measure, Integration & Real Analysis · Problem 2B.25

Measures — Sheldon Axler

Mathematics — Measure Theory · monotone-functions

Problem

Suppose

B \subseteq \mathbf{R}

and

f : B \to \mathbf{R}

is an increasing function. Prove that there exists a sequence

f_1, f_2, \ldots

of strictly increasing functions from

B

\mathbf{R}

such that

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

for every

x \in B