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](https://planetmath.org/proofofexistenceoftheessentialsupremum), 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?




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This is a cross-posting from [this question on math.stackexchange](https://math.stackexchange.com/questions/304696/from-point-wise-to-essential-supremum-of-a-set-of-real-valued-measurable-functio) ([Wayback Machine](https://web.archive.org/web/20151024010145/http://math.stackexchange.com/questions/304696/from-point-wise-to-essential-supremum-of-a-set-of-real-valued-measurable-functio))




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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](https://planetmath.org/essentialsupremum).


The indexed intersection is defined as usual: $\bigcap\_{i\in I}S\_i = \{x|\forall i\in I: x\in S\_i\}$



From point-wise to essential supremum of a set of real-valued measurable functions
Okay. Give $[0,1]$ Lebesgue measure. For each $t \in [0,1]$ let $S\_t = \{1\_{\{t\}}\} \cup \{a\cdot 1\_{[0,1]}: a \geq 1\}$.


Then $\bigcap S\_t = \{a\cdot 1\_{[0,1]}: a \geq 1\}$ and its inf is the function $1\_{[0,1]}$. For each $t$ the inf of $S\_t$ is the function $1\_{\{t\}}$, and the sup of these is also the function $1\_{[0,1]}$. So your premise holds.


The essential inf of $\bigcap S\_t$ is also the function $1\_{[0,1]}$ but the essential inf of each $S\_t$ is the zero function and their essential sup is again the zero function. So the conclusion fails.


Is this really what you meant?


