Let $U$ be uniformly distributed on the interval $ [0, 1] $. Consider the decimal digits of $ U $, calling them $ Z_1, Z_2, Z_3, \ldots $ Consider $C =\bigcup σ(Z_1, Z_2, \ldots , Z_n) $. This is an increasing union of σ-algebras. a) Is $C $ a σ-algebra? b)$C$ is as large as $σ(U) $,
I am looking for some counterexample for part (a) and what are the examples of sets that are in $σ(U)$ but not in $C$ with valid convincing argument.