Measure, Integration & Real Analysis · Problem 2B.29

Measures — Sheldon Axler

Mathematics — Measure Theory · measurable-functions

Problem

Give an example of a measurable space

(X, \mathcal{S})

and a family

\{f_t\}_{t \in \mathbf{R}}

such that each

f_t

is an

\mathcal{S}

-measurable function from

X

[0,1]

, but the function

f : X \to [0,1]

defined by

f(x) = \sup\{f_t(x) : t \in \mathbf{R}\}

is not

\mathcal{S}

-measurable. [Compare this exercise to 2.53, where the index set is

\mathbf{Z}^+

rather than

\mathbf{R}