In
mathematics, more specifically
measure theory, there are various notions of the
convergence of measures. For an intuitive general sense of what is meant by
convergence in measure, consider a sequence of measures µ
n on a space, sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better and better' approximations to a desired measure µ that is difficult to obtain directly. The meaning of 'better and better' is subject to all the usual caveats for taking
limits; for any error tolerance e > 0 we require there be
N sufficiently large for
n =
N to ensure the 'difference' between µ
n and µ is smaller than e. Various notions of convergence specify precisely what the word 'difference' should mean in that description; these notions are not equivalent to one another, and vary in strength.