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 supφ∈Φ φ(x) ≡ 1 whereas φ = 0 P-a.s. for each single φ ∈ Φ. This suggests the following notion of an essential supremum defined in terms of almost sure inequalities.

Theorem A.37. Let Φ be any set of random variables on (Ω,F, P).

(a) There exists a random variable φ∗ with the following two properties.

(i) φ∗ ≥ φ P-a.s. for all φ ∈ Φ.

(ii) φ∗ ≤ ψ P-a.s. for every random variable ψ satisfying ψ ≥ φ P-a.s. for all φ ∈ Φ.

(b) Suppose in addition that Φ is directed upward, i.e., for φ, ∈ Φ there exists ψ ∈ Φ with ψ ≥ φ ∨ . Then there exists an increasing sequence φ1 ≤ φ2 ≤· · · in Φ such that φ∗ = limn φn P-almost surely.

Definition A.38. The random variable φ∗ in Theorem A.37 is called the essential supremum of Φ with respect to P, and we write

The essential infimum of Φ with respect to P is defined as

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Proof of Theorem A.37. Without loss of generality, we may assume that each φ ∈ Φ takes values in [0, 1]; otherwise we may consider := {f ◦ φ | φ ∈ Φ} with f : ℝ → [0, 1] strictly increasing.

If Ψ ⊂ Φ is countable, let φΨ(ω) := supφ∈Ψ φ(ω). Then φΨ is measurable. We claim that the upper bound

c := sup{E[ φΨ ]| Ψ ⊂ Φ countable}

is attained by some countable Ψ∗ ⊂ Φ. To see this, take Ψn with E[ φΨn ] → c and let Then Ψ∗ is countable and E[ φΨ∗ ] = c.

We now show that φ∗ := φΨ∗ satisfies (i). Suppose that (i) does not hold. Then there exists φ ∈ Φ such that P[ φ > φ∗] > 0. Hence Ψʹ := Ψ∗ ∪ {φ} satisfies

E[ φΨ] > E[ φΨ∗ ] = c,

in contradiction to the definition of c. Furthermore, if ψ is any other random variable satisfying ψ ≥ φ P-a.s. for all φ ∈ Φ, then obviously ψ ≥ φ∗.

Finally, the construction shows that φΨ∗ can be approximated by an increasing sequence if Ψ is directed upwards.

Remark A.39. For a given random variable X let Φ be the set of all constants c such that P[ X > c ] > 0. The number

ess sup X := sup Φ

is the smallest constant c ≤ +∞ such that X ≤ c P-a.s. and called the essential supre-mum of X with respect to P. The essential infimum of X is defined as

ess inf X := ess sup(−X).

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