Almost Sure

6 January 19

Essential Suprema

Filed under: Probability Theory — George Lowther @ 2:00 PM
Tags: ,

Given a sequence {X_1,X_2,\ldots} of real-valued random variables defined on a probability space {(\Omega,\mathcal F,{\mathbb P})}, it is a standard result that the supremum

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle X\colon\Omega\rightarrow{\mathbb R}\cup\{\infty\},\smallskip\\ &\displaystyle X(\omega)=\sup_nX_n(\omega). \end{array}

is measurable. To ensure that this is well-defined, we need to allow X to have values in {{\mathbb R}\cup\{\infty\}}, so that {X(\omega)=\infty} whenever the sequence {X_n(\omega)} is unbounded above. The proof of this fact is simple. We just need to show that {X^{-1}((-\infty,a])} is in {\mathcal F} for all {a\in{\mathbb R}}. Writing,

\displaystyle  X^{-1}((-\infty,a])=\bigcap_nX_n^{-1}((-\infty,a]),

the properties that {X_n} are measurable and that the sigma-algebra {\mathcal F} is closed under countable intersections gives the result.

The measurability of the suprema of sequences of random variables is a vital property, used throughout probability theory. However, once we start looking at uncountable collections of random variables things get more complicated. Given a, possibly uncountable, collection of random variables {\mathcal S}, the supremum {S=\sup\mathcal S} is,

\displaystyle  S(\omega)=\sup\left\{X(\omega)\colon X\in\mathcal S\right\}. (1)

However, there are a couple of reasons why this is often not a useful construction:

  • The supremum need not be measurable. For example, consider the probability space {\Omega=[0,1]} with {\mathcal F} the collection of Borel or Lebesgue subsets of {\Omega}, and {{\mathbb P}} the standard Lebesgue measure. For any {a\in[0,1]} define the random variable {X_a(\omega)=1_{\{\omega=a\}}} and, for a subset A of {[0,1]}, consider the collection of random variables {\mathcal S=\{X_a\colon a\in A\}}. Its supremum is

    \displaystyle  S(\omega)=1_{\{\omega\in A\}}

    which is not measurable if A is a non-measurable set (e.g., a Vitali set).

  • Even if the supremum is measurable, it might not be a useful quantity. Letting {X_a} be the random variables on {(\Omega,\mathcal F,{\mathbb P})} constructed above, consider {\mathcal S=\{X_a\colon a\in[0,1]\}}. Its supremum is the constant function {S=1}. As every {X\in\mathcal S} is almost surely equal to 0, it is almost surely bounded above by the constant function {Y=0}. So, the supremum {S=1} is larger than we may expect, and is not what we want in many cases.

The essential supremum can be used to correct these deficiencies, and has been important in several places in my notes. See, for example, the proof of the debut theorem for right-continuous processes. So, I am posting this to use as a reference. Note that there is an alternative use of the term `essential supremum’ to refer to the smallest real number almost surely bounding a specified random variable, which is the one referred to by Wikipedia. This is different from the use here, where we look at a collection of random variables and the essential supremum is itself a random variable.

The essential supremum is really just the supremum taken within the equivalence classes of random variables under the almost sure ordering. Consider the equivalence relation {X\cong Y} if and only if {X=Y} almost surely. Writing {[X]} for the equivalence class of X, we can consider the ordering given by {[X]\le[Y]} if {X\le Y} almost surely. Then, the equivalence class of the essential supremum of a collection {\mathcal S} of random variables is the supremum of the equivalence classes of the elements of {\mathcal S}. In order to avoid issues with unbounded sets, we consider random variables taking values in the extended reals {\bar{\mathbb R}={\mathbb R}\cup\{\pm\infty\}}.

Definition 1 An essential supremum of a collection {\mathcal S} of {\bar{\mathbb R}}-valued random variables,

\displaystyle  S = {\rm ess\,sup\,}\mathcal{S}

is the least upper bound of {\mathcal{S}}, using the almost-sure ordering on random variables. That is, S is an {\bar{\mathbb R}}-valued random variable satisfying

  • upper bound: {S\ge X} almost surely, for all {X\in\mathcal S}.
  • minimality: for all {\bar{\mathbb R}}-valued random variables Y satisfying {Y\ge X} almost surely for all {X\in\mathcal S}, we have {Y\ge S} almost surely.

(more…)

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