Measure, Integration & Real Analysis · Problem 12.17

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · trigonometric-family-independence

Problem

Suppose

\mathcal{B}

is the

\sigma

-algebra of Borel subsets of

(-\pi, \pi]

and

P

is Lebesgue measure on

\bigl((-\pi, \pi], \mathcal{B}\bigr)

divided by

2\pi

. Let

\{e_k\}_{k \in \mathbf{Z} \setminus \{0\}}

be the family of trigonometric functions defined by the third bullet point of Example 8.51 (notice that

k = 0

is excluded). (a) Show that

\{e_k\}_{k \in \mathbf{Z} \setminus \{0\}}

is not an independent family of random variables. (b) Show that

\{e_k\}_{k \in \mathbf{Z} \setminus \{0\}}

is an identically distributed family.