# From point-wise to essential supremum of a set of real-valued measurable functions

I want to prove some property about essential suprema and I think I can show them for the pointwise supremum $\sup S$.

The problem is, that the sets involved are uncountable and thus, the point-wise suprema need not be measurable.

Is there a way to transfer the results from the point-wise to the essential supremum?

I know the following does not make sense, since $\sup S$ might not be measurable, but is there something comparable?

$\mathrm{ess}\sup S=\sup S \;\;$ almost surely

If it helps, the statement I would could prove for an uncountable collectino of uncountable sets of random functions $\{S_i\}_{i\in I}$:

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

Now I want to show

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

I cross posted to Mathoverflow

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What is the essential supremum of a set? – Michael Greinecker Feb 15 '13 at 18:55
it is defined for a set of real-valued measurable function. (see planetmath.org/EssentialSupremum.html) – JSG Feb 15 '13 at 19:25
So why are you taking intersections? – Michael Greinecker Feb 15 '13 at 19:27
The intersection of a collection of sets of functions. Sorry, if it is not clear. – JSG Feb 16 '13 at 10:07