I’ve been brushing up on my functional analysis as part of my broader study of probability theory by reading Adam Bobrowski’s text Functional Analysis for Probability and Stochastic Processes. Over the past couple days, I’ve been focused on the topic of Markov operators, which I would like to discuss at some length in today’s post. We will later use an extension theorem for Markov operators to prove the existence of conditional expectation for absolutely integrable random variables.
Let be a measure space, and let be a (not necessarily closed) subspace of which is dense in and has the following property: implies that . Note that this requirement is the same as implies , since .
Suppose is a linear operator such that and , for all . Then there exists a unique extension of to a bounded linear operator on and for all in .
Proof. For any , we write , where and . For any , and in , hence . Since by hypothesis,
Noting that , we have
We see that is a bounded linear operator on with . Since is dense in , has a unique extension to a bounded linear operator defined on the entire space . We abuse notation and denote the extension also by .
To see that maps nonnegative to , fix and choose a sequence in such that as . Since
we see that . Thus, .
Such a subspace exists, since by definition of the Lebesgue integral, we can always take to be the subspace of simple functions. If is a finite measure, then we can take to be the space .
The above result leads us to define a class of operators on , for a given measure space . A linear operator is said to be a Markov operator if
- for all ;
- for .
Note that the second condition (together with the extension theorem) implies that preserves the integral of , for all , since
Thus, a Markov operator has operator norm .
We now give some examples of Markov operators. Let our measure space be with the Lebesgue measure and -algebra, and for nonnegative such that , define
It is evident that is linear. If , then
since for all and fixed. Lastly, by Fubini’s theorem and translation invariance,
Thus, is a Markov operator.
Now consider a measure space and a measurable space , and let be a measurable map. We seek a measure on such that the operator
is a Markov operator. I claim the desired measure is given by the pushforward measure . Indeed, for ,
by the change-of-variables theorem.
We now extend the definition of Markov operators to the case where maps equivalence classes of nonnegative integrable functions on the measure space to equivalence classes of nonnegative integrable functions on a possibly distinct measure space and
for all nonnegative .
Let be a probability space on which a countable collection of i.i.d. random variables. For each , define a random variable . The reader can verify that . For any , define
Let denote the Lebesgue measure. I claim that is a Markov operator with domain . Suppose are nonnegative and except on a set of measure zero. Let denote the set of such that . Since the law of is absolutely continuous with respect to Lebesgue measure, for each . Hence, is an event with probability zero. For any , . Hence, is an event of probability zero.
For nonnegative , we have by the monotone convergence theorem and change-of-variables theorem that