A.5 The essential supremum of a family of random variables
In this section, we discuss the essential supremum of an arbitrary family Φ of random variables on a given probability space (Ω,F, P). Consider first the case in which the set Φ is countable. Then φ∗(ω) := supφ∈φ(ω) will also be a random variable, Φ i.e., φ∗ is measurable. Measurability of the pointwise supremum, however, is not guaranteed if Φ is uncountable. Even if the pointwise supremum is measurable, it may not be the right concept, when we focus on almost sure properties. This can be illustrated by taking P as the Lebesgue measure on Ω := [0, 1] and Φ := {
{x} | 0 ≤ x ≤ 1}. Then sup ...
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