Convergence of Measures - Informal Descriptions

Informal Descriptions

This section attempts to provide a rough intuitive description of three notions of convergence, using terminology developed in calculus courses; this section is necessarily imprecise as well as inexact, and the reader should refer to the formal clarifications in subsequent sections. In particular, the descriptions here do not address the possibility that or could be infinite or zero.

The various notions of weak convergence formalize the assertion that the 'average value' of each 'nice' function should converge:

To formalize this requires a careful specification of the set of functions under consideration, the domain of integration, and of what is meant by average value (see expectation). This notion treats convergence for different functions independently of one another, i.e. different functions may require different values of to be approximated equally well (thus, a form of non-uniform convergence).

The notion of strong convergence formalizes the assertion that the measure of each measurable set should converge:

Intuitively, considering integrals of 'nice' functions, this notion provides more uniformity than weak convergence: for any and any upper bound, we can find an so that ensures

for all integrable functions with bounded by . (The definition of strong convergence does not require one to specify a set of measurable functions, so the interpretation given here might not apply.) This notion of convergence still allows to vary with the set .

The notion of total variation convergence formalizes the assertion that the measure of all measurable sets should converge uniformly, i.e. we do not allow the described above to depend upon which set we are measuring. (This is only a very rough description of total variation convergence, as additional technical care is necessary; see below.)