A.3.1 Probability spaces

A probability space is given by

1. a set encoding all possible random outcomes,
2. a -field , which is a set constituted of certain subsets of , and satisfies
   • the set is in ,
   • if is in , then its complement is in ,
   • if for in is in , then is in ,
3. a probability measure , which is a mapping satisfying
   • it holds that ,
   • the -additivity property: if for in are pairwise disjoint sets of , then
     

The elements of are called events, and regroup certain random outcomes leading to situations of interest in such a way that these can be attributed a “likelihood measure” using .

Clearly, , and and more generally , and if for in is in , then . Note that in order to consider finite unions or intersections it suffices to use or where necessary.

The trivial -field is included in any -field, which is in turn included in the -field of all subsets of . The latter is often the one of choice when possible, and notably if is countable, but it is often too large to define an appropriate probability measure on it. Moreover, the notion of sub--field is used to encode partial information available in a probabilistic model.

The following important property is in fact equivalent to -additivity, using the fact that is a finite measure.

A.3.2 Measurable spaces and functions: signed and nonnegative

A set furnished with a -field is said to be measurable. A mapping from a measurable set with -field to another measurable set with -field is said to be measurable if and only if

A (nonnegative) measure on a measurable set with -field is a -additive mapping .

By -additivity, if and are in and , then . The measure is said to be finite if , and then , and to be a probability measure or a law if , and then .

Many results for probability spaces can be extended in this framework (which is usually introduced first) using the classical computation conventions in .

For instance, if this quantity has a meaning. As in Lemma A.3.1, the -additivity property is equivalent to the fact that if is a nondecreasing sequence of events in , then . Moreover, by complementation, if is a nonincreasing sequence of events in s.t. for some , then .

A further extension is given by signed measures , which are -additive mappings . The Hahn–Banach decomposition yields an essentially unique decomposition of a signed measure into a difference of nonnegative finite measures, under the form , in which the supports and of and are disjoint. The finite nonnegative measure is called the total variation measure of , and its total mass is called the total variation norm of .

The space of all signed measures is a Banach space for this norm, which can be identified with a closed subspace of the strong dual of the functional space .

For every (nonnegative, possible infinite) reference measure , the Banach space contains a subspace that can be identified with by identifying any measure , which is absolutely continuous w.r.t. with its Radon–Nikodym derivative . If is discrete, then a natural and universal choice for is the counting measure, and thus can be identified with the collection and with .

A.3.3 Random variables, their laws, and expectations

A.3.3.1 Random variables and their laws

A probability space is given. A random variable (r.v.) with values in a measurable set with -field is a measurable function , which satisfies

For an arbitrary mapping , the set

is a -field, called the -field generated by , encoding the information available on by observing . Notably, is measurable if and only if .

The probability space is often only assumed to be fixed without further precision and represents some kind of ideal probabilistic knowledge. Only the properties of certain random variables are precisely given. These often represent indirect observations or effects of the random outcomes, and it is natural to focus on them to get useful information.

The law of the r.v. is the probability measure on , which is well defined as is measurable. It is denoted by or and is given by

Then, is a probability space which encodes the probabilistic information available on the outcomes of .

A.3.3.2 Expectation for -valued random variables

The expectation will be defined as a monotone linear extension of the probability measure , first for random variables taking a finite number of values in , then for general random variables with values in , and finally for real random variables satisfying an integrability condition. The notation is sometimes used to stress .

This procedure allows to define the integral of a measurable function , from with -field to with -field , by a measure , but we restrict this to probability measures for the sake of concision.

The classic structure of is extended to by setting

Finite number of values

If is an r.v. taking a finite number of values in , then

In particular,

For such random variables, this defines a monotone operator, in the sense that

which moreover is nonnegative linear, in the sense that

Extension by supremum

For an r.v. with values in , let

and

This extension of is still monotone and nonnegative, from which we deduce the following extension of the monotone limit lemma (Lemma A.3.1). This is where the fact that is measurable becomes crucial.

Nonnegative linearity

This theorem allows to prove that is nonnegative linear, by replacing the supremum in the definition by the limit of an adequate nondecreasing sequence. If is a -valued r.v., then we define for the dyadic approximation satisfying

If and are -r.v., and , then

Fatou's Lemma

An important corollary of the monotone convergence theorem is the following.

Let us finish with a quite useful result.

A.3.3.3 Real-valued random variables and integrability

Let be an r.v. with values in . Let

so that

The natural extension to of the operations on lead to setting, except if the indeterminacy occurs,

This definition is monotone and linear: if all is well defined in , then

A.3.3.4 Integrable random variables

In particular,

and the latter is well defined. This is the most useful case and is extended by linearity to define for with values in satisfying for some (and then every) norm . Then, is said to be integrable. The integrable random variables form a vector space

It is a simple matter to check that if is an r.v. with values in and is measurable then

in all cases in which one of these expressions can be defined, and then all can.

The expectation has good properties w.r.t. the a.s. convergence of random variables. The monotone convergence theorem has already been seen. Its corollary the Fatou lemma will be used to prove an important result.

A sequence of -valued random variables is said to be dominated by an r.v. if

and to be dominated in by if moreover . The sequence is thus dominated in if and only if

This theorem can be extended to the case when

Indeed, the Borel–Cantelli lemma implies that then, from each subsequence, a further subsubsequence converging a.s. can be extracted. Applying Theorem A.3.5 to this a.s. converging sequence yields that the only accumulation point in for is , and hence that .

A.3.3.5 Convexity inequalities and spaces

For , we will check that the set of all -valued random variables s.t. forms a Banach space, denoted by

if two a.s. equal random variables are identified (i.e., on the quotient space). In particular, is a Hilbert space with scalar product

This proof remains valid if is replaced by an arbitrary positive measure. The case is a special case of the Cauchy–Schwarz inequality.

The Jensen inequality yields that if , then . The linear form hence has operator norm , as

with equality for constant .

A.3.4 Random sequences and Kolmogorov extension theorem

Let us go back to Section 1.1. Let be given a family of laws on , for and in . Two natural questions arise:

• Does there exist a probability space , a -field on , and an r.v. , satisfying that
  

  that is, this family of laws are the finite-dimensional marginals of ?

• If it is so, is the law of unique, that is, is it characterized by its finite-dimensional marginals?

Clearly, the must be consistent, or compatible: if is a -tuple included in the -tuple , then must be equal to the corresponding marginal of .

It is natural and “economical” to take , called the canonical space, the process given by the canonical projections

called the canonical process, and to furnish with the smallest -field s.t. each and hence each is an r.v.: the product -field

Note that if is a sequence of subsets of the discrete space , then

and that events of this form are sufficient to characterize convergence in results such as the pointwise ergodic theorem (Theorem 4.1.1). See also Section 2.1.1.

By construction, is measurable and hence an r.v. on furnished with the product -field, and if this space is furnished with a probability measure , then has law .

The following result is fundamental. It is relatively easy to show the uniqueness part: any two laws on the product -field with the same finite-dimensional marginals are equal. The difficult part is the existence result, which relies on the Caratheodory extension theorem.

The explicit form given in Definition 1.2.1, in terms of the initial law and the transition matrix , allows to check easily that these probability measures are consistent. The Kolmogorov extension theorem then yields the existence and uniqueness of the law of the Markov chain on the product space. This yields the mathematical foundation for all the theory of Markov chains.