Measure, Integration & Real Analysis · Problem 9A.11

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · dirac-delta-measure-norm

Problem

For

(X, \mathcal{S})

a measurable space and

b \in X

, define a finite (positive) measure

\delta_b

(X, \mathcal{S})

\delta_b(E) = \begin{cases} 1 & \text{if } b \in E, \\ 0 & \text{if } b \notin E \end{cases}

for

E \in \mathcal{S}

. (a) Show that if

b, c \in X

, then

\|\delta_b + \delta_c\| = 2

. (b) Give an example of a measurable space

(X, \mathcal{S})

and

b, c \in X

with

b \neq c

such that

\|\delta_b - \delta_c\| \neq 2