Measure, Integration & Real Analysis · Problem 2C.8

Measures — Sheldon Axler

Mathematics — Measure Theory · measures

Problem

Give an example of a set

X

, a

\sigma

-algebra

\mathcal{S}

of subsets of

X

, a set

\mathcal{A}

of subsets of

X

such that the smallest

\sigma

-algebra on

X

containing

\mathcal{A}

is

\mathcal{S}

, and two measures

\mu

and

\nu

on

(X, \mathcal{S})

such that

\mu(A) = \nu(A)

for all

A \in \mathcal{A}

and

\mu(X) = \nu(X) < \infty

, but

\mu \neq \nu

.

Browse the problem library