Apologies for the poor formatting, not sure how to get type piece-wise function on stack.
Given a symmetric random walk $X_n$, what does it mean to say that $X_{n+1}=X_n+1$ with $E(\cdot |F_n)=\frac{1}{2}$ and $X_{n+1}= X_n-1$ with $E( \cdot |F_n) = \frac{1}{2}$? Where $F_n$ is the natural filtration. As opposed to saying $\mathbb{P}(X_{n+1}=X_n+1) = \frac{1}{2}$ and $\mathbb{P}(X_{n+1}=X_n -1 ) =\frac{1}{2}$. I supposed what I'm trying to understand is how to intuitively view conditional expectation on a sigma algebra.