To show that a given set is a Borel set

Question

How can I show that the set $ E_n = \{ x | 1/n \hash\{i = 1, 2, \ldots , n | x_i = 7\} \geq \alpha \} $ is a Borel set for all real $\alpha$ where $x∈(0,1)$ has the decimal expansion $0.x_1 x_2 x_3 \dots$

Answer 1

Denote the $i$-th digit map $(0,1) \to \{0,\ldots, 9\}$ by $\pi_i(x) = x_i$. Note that $$ B_i := \{x_i = 7\} = \pi_i^{-1}(7) = \bigcup_{k=0}^{10^{i-1}} \bigl(k\cdot 10^{1-i} + [7\cdot 10^{-i}, 8\cdot 10^{-i})\bigr) $$ is a Borel set as a finite union of intervals. Hence $\chi_{B_i}$, the characteristic function of $B_i$ is a Borel function, therefore, the linear combination $\phi_n := \frac 1n \sum_{i=1}^n \chi_{B_i}$ also is a Borel function. Hence $$ E_n := \{\phi_n \ge \alpha \}$$ is a Borel set.