In my notes we have $\mathbb{E}[\hat{Y}_n\mid\mathcal{F}_t]=X_t$ where $\hat{Y}_n=(Y^1_T+\cdots+Y^n_T)/n$ and $\mathbb{E}[Y^i_T \mid \mathcal{F}_t]=X_t$. We say that $\hat{Y}_n$ is unbiased, but I thought being unbiased meant the expectation of the r.v. equals the quantity being estimated, not conditional expectation. Similarly for the SLLN. For example, here they write the SLLN in a conditional form, but even then they seem to define a conditional r.v. which isn't good practice? https://quant.stackexchange.com/questions/2636/simulating-conditional-expectations Is it fine to use conditional expectations in these defintions / theorems?
$\textbf{Edit:}$ We take $t<T$ and $n$ to be the number of realisations of $(Y_T^i)_{i\in \mathbb{Z}}$. I consider $\mathcal{F}_t$ as a filtration for my process $(Y_t)$ and want to look at the special case of the process $Y_T$ at some end time $T$ for multiple realizations (my aim is to simulate the process).