I need help finding a conditional expectation:
Let $X$ be a $(0,1)$ uniform random variable i.e. $\mathbb{P}(X \in A)=\lambda((0,1)\cap A)$ where $\lambda$ is the Lebuesgue measure. We define the random variables $$X_n = \frac{\left\lfloor10^nX \right\rfloor}{10^n}$$
and we need to find $$\mathbb{E}(h(X_{n+1})|X_n) \text{ and }\mathbb{E}(h(X)|X_n)$$ where $h\colon [0,1]\to \mathbb{R}$ is some bounded measurable function.
I noticed that $X_n$ has only $n$ decimals of $X$ hence $\sigma(X_n) \subseteq \sigma(X_{n+1})$ and that $X_n \to X$ a.s, but I don't know if this is of any help.
Any hint, reference or solution will be appreciated :)!
To quote Durrett, 'the game is "guess and verify."'.
Note that if you know $X_n$, then you must have $X_{n+1} = X_n + { 1\over 10^{(n+1)} } d$, where $d \in \{0,...,9\}$ and each $d$ is equally likely.
If you know $X_n$, then you must have $X=X_n+ { 1\over 10^{n} } u$, where $u \in [0,1)$, where $u$ is uniformly distributed.