Measure, Integration & Real Analysis · Problem 9B.7

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · cantor-set-singular-measure

Problem

Use the Cantor set to prove that there exists a (positive) measure

\nu

(\mathbf{R}, \mathcal{B})

such that

\nu \perp \lambda

and

\nu(\mathbf{R}) \neq 0

but

\nu(\{x\}) = 0

for every

x \in \mathbf{R}

; here

\lambda

denotes Lebesgue measure on the

\sigma

-algebra

\mathcal{B}

of Borel subsets of

\mathbf{R}

. [The second bullet point in Example 9.29 does not provide an example of the desired behavior because in that example,

\nu(\{r_k\}) \neq 0

for all

k \in \mathbf{Z}^+

with

w_k \neq 0