For an uncountable collection of uncountable sets of real-valued random variables (i.e. measurable with respect to a $\sigma$-algebra) $\{S_i\}_{i\in I}$, with

$\inf \left(\bigcap_{i\in I}S_i \right)= \sup\{\inf S_i|{i\in I}\}$

I want to to show

$\mathrm{ess}\inf \left(\bigcap_{i\in I}S_i \right)= \mathrm{ess}\sup\{\mathrm{ess}\inf S_i|{i\in I}\}$

I tried something in analogy to the proof of existence of the essential supremum, but failed.

It would be great to get some help on this. Does it hold and if so, how prove it and if not, why not?

This is a cross-posting from this question on math.stackexchange

Edit: I change the notation. No the all infima/suprema are understood to be pointwise suprema of a set of functions. And the essential suprema/infima of a set of functions should be read like on this page.

The indexed intersection is defined as usual: $\bigcap_{i\in I}S_i = \{x|\forall i\in I: x\in S_i\}$