Measure, Integration & Real Analysis · Problem 2B.13

Measures — Sheldon Axler

Mathematics — Measure Theory · measurable-function-supremum-sequence

Problem

Verify the equation

h^{-1}\bigl((a, \infty]\bigr) = \bigcup_{k=1}^\infty f_k^{-1}\bigl((a, \infty]\bigr)

used in the proof of result 2.53, where

h(x) = \sup\{f_k(x) : k \in \mathbf{Z}^+\}