Question:
Suppose we have an integrable random variable $X:\Omega\to\mathbb{R}$. Denote partition of $\mathbb{R}$ by $\Pi^n = \{\Pi^n (k) \subset \mathbb{R} : k = 1,\dots, n\}$ and the corresponding generated sigma-algebra $\sigma^{n} = \sigma(\{X \in \Pi^n(k)\}, k=1,\dots,n)$. Assume $\Pi^{n+1}$ is a refinement of $\Pi^n$ i.e. every component in $\Pi^{n+1}$ is a subset of a component in $\Pi^n$.
Define random variable $X^n:= E[X\mid\sigma^n]$. For a fixed sample path $w \in \Omega$, is the set $\{ X^n (w)\}_{n\in\mathbb{N}}$ bounded?
My thoughts:
I recently asked a similar question here which was answered. However, the counterexample provided involved independently and identically copies of the random variable. So I was curious if the set is now bounded if we condition the same variable on finer and finer partitions.
I'm going to answer my own question. I hope this is right and helps someone in the future.
I'm going to assume that $\sigma(X) = \sigma(\cup_{n\in\mathbb{N}} \sigma^n)$. Then we know that $X^n$ converges to $X$ with probability one. Therefore, for all paths $w$ from the set of non-zero probability paths, the sequence $(X^n(w))_{n\in\mathbb{N}}$ converges and is therefore bounded.