Measure, Integration & Real Analysis · Problem 9A.8

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · total-variation-norm-complex-measure

Problem

Suppose

t > 0

and

\lambda

is Lebesgue measure on the

\sigma

-algebra of Borel subsets of

[0, t]

. Suppose

h: [0, t] \to \mathbf{C}

is the function defined by

h(x) = \cos x + i \sin x.

Let

\nu

be the complex measure defined by

d\nu = h\, d\lambda

. (a) Show that

\|\nu\| = t

. (b) Show that if

E_1, E_2, \ldots

is a sequence of disjoint Borel subsets of

[0, t]

, then

\sum_{k=1}^{\infty} |\nu(E_k)| < t.

[This exercise shows that the supremum in the definition of

|\nu|([0, t])

is not attained, even if countably many disjoint sets are allowed.]