Measure, Integration & Real Analysis · Problem 12.16

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · iid-family-haar-functions

Problem

Suppose

\mathcal{B}

is the

\sigma

-algebra of Borel subsets of

[0,1)

and

P

is Lebesgue measure on

([0,1), \mathcal{B})

. Let

\{e_k\}_{k \in \mathbf{Z}^+}

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

k = 0

is excluded). Show that the family

\{e_k\}_{k \in \mathbf{Z}^+}

is an i.i.d.