In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. The definition used in measure theory is closely related to, but not identical to, the definition typically used in probability.
Let be a positive measure space. A set is called uniformly integrable if to each there corresponds a such that
- A class of random variables is called uniformly integrable (UI) if given , there exists such that , where is the indicator function
- An alternative definition involving two clauses may be presented as follows: A class of random variables is called uniformly integrable if:
- There exists a finite such that, for every in , and
- For every there exists such that, for every measurable such that and every in , .
The two probabilistic definitions are equivalent.
Relationship between definitionsEdit
The two definitions are closely related. A probability space is a measure space with total measure 1. A random variable is a real-valued measurable function on this space, and the expectation of a random variable is defined as the integral of this function with respect to the probability measure. Specifically,
Let be a probability space. Let the random variable be a real-valued -measurable function. Then the expectation of is defined by
Then the alternative probabilistic definition above can be rewritten in measure theoretic terms as: A set of real-valued functions is called uniformly integrable if:
- There exists a finite such that, for every in , .
- For every there exists such that, for every measurable such that and for every in , .
Comparison of this definition with the measure theoretic definition given above shows that the measure theoretic definition requires only that each function be in . In other words, is finite for each , but there is not necessarily an upper bound to the values of these integrals. In contrast, the probabilistic definition requires that the integrals have an upper bound.
One consequence of this is that uniformly integrable random variables (under the probabilistic definition) are tight. That is, for each , there exists such that
In contrast, uniformly integrable functions (under the measure theoretic definition) are not necessarily tight.
In his book, Bass uses the term uniformly absolutely continuous to refer to sets of random variables (or functions) which satisfy the second clause of the alternative definition. However, this definition does not require each of the functions to have a finite integral. The term "uniform absolute continuity" is not standard, but is used by some other authors.
The following results apply to the probabilistic definition.
- Definition 1 could be rewritten by taking the limits as
- A non-UI sequence. Let , and define
- By using Definition 2 in the above example, it can be seen that the first clause is satisfied as norm of all s are 1 i.e., bounded. But the second clause does not hold as given any positive, there is an interval with measure less than and for all .
- If is a UI random variable, by splitting
- If any sequence of random variables is dominated by an integrable, non-negative : that is, for all ω and n,
- A class of random variables bounded in ( ) is uniformly integrable.
In the following we use the probabilistic framework, but regardless of the finiteness of the measure, by adding the boundedness condition on the chosen subset of .
Relation to convergence of random variablesEdit
A sequence converges to in the norm if and only if it converges in measure to and it is uniformly integrable. In probability terms, a sequence of random variables converging in probability also converge in the mean if and only if they are uniformly integrable. This is a generalization of Lebesgue's dominated convergence theorem, see Vitali convergence theorem.
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