I read an answer from this question: $E[e_te_s\Delta B_t\Delta B_s]$ for $\Delta B_t$ Brownian motion increments and $e_t(\omega)$ a measurable function.
I don't quite understand the meaning of $\mathcal{F}$-measurable, even though I have read its definition and this answer: What does it mean by $\mathcal{F}$-measurable?
Is there any intuitive explanation on what is $\mathcal{F}$-measurable, and why it is obvious that $\mathcal{F}_{t_i}\subset \mathcal{F}_{t_j}$ if $i<j$? How to decide whether a random variable is $\mathcal{F}$-measurable or not?
