Almost surely

The weird locution almost surely is to probability theory as almost everywhere is to measure theory. Imagine throwing a dart at the unit square; the probability that the dart lands in any subregion of the square is the area of that subregion. The area of the diagonal of the square is zero, so the probability that the dart lands exactly on the diagonal is zero. But the diagonal is not the empty set; a point on the diagonal is no less probable than is any point at which the dart could land. One says that the dart will almost surely not land on the diagonal. In other words an event is "almost sure" if the probability of its complement is zero, even though it may not be empty. The phrase appears in speaking of almost sure convergence of random variables.

An example of the fine distinction between 'sure' and 'almost sure' can be found in the difference between constant and almost surely constant random variables. In measure theoretic probability theory these two types of random variable are not identical, but for practical purposes they are equivalent, since if a constant random variable  X  and an almost surely constant random variable  Y  represent ths same constant  c, then they share the same distribution functions.