I feel kinda stupid for asking this, but anyway, I have a quick questionabout Kolmogorov's Maximal Inequality, which can be stated as follows:
Let $ (X_n)_{n \in \mathcal{Z}} $ be a sequence of martingales, then:
$P\left(\max_{1 \leq t \leq n} |X_t| \geq \delta\right)\leq \frac{1}{\delta^2}E[X_n^2] $
My question is, since we are dealing with a finite number of random variables, can the inequality be rewritten using the supremum?
For example, like this:
$P\left(\sup_{1 \leq t \leq n} |X_t| \geq \delta\right)\leq \frac{1}{\delta^2}E[X_n^2] $
My argument is simply that for any fixed $ \omega \in \Omega$, we will have that $ \max_n X_n(w)=\sup_n X_n $ because we are dealing with a finite set of reals.
I suspect that I am missing something really obvious, so I hope someone will correct me because I have never seen the inequality written with the supremum.
Thanks!