Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}, \quad F_n(x) = \mathbb{P}(Y_n \leqslant x), \quad G_n(x) = \mathbb{P}(Y_n < x).$$ We denote $D = \bigcup_{n \in \mathbb{N}^{\star}} D_n$ where $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$. Generalize the results obtained in the previous question for all $x \in [0,1]$.
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set
$$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}, \quad F_n(x) = \mathbb{P}(Y_n \leqslant x), \quad G_n(x) = \mathbb{P}(Y_n < x).$$
We denote $D = \bigcup_{n \in \mathbb{N}^{\star}} D_n$ where $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$.
Generalize the results obtained in the previous question for all $x \in [0,1]$.