A measure

Measure is a way of gauging how big something is – in terms of length, volume, or some other quality. One of the strangest facts in mathematics is that some objects exist that can't be measured. In the language of sets, the basic rules (somewhat simplified) of mathematical measures are as follows: (1) the measure of any set is a real number; (2) the empty set has measure zero; (3) if A and B are two sets with no elements in common then the measure of A the union of A and B is equal to the measure of A plus the measure of B. The second of these rules can be very useful, for example, when integrating a function, since it allows us to ignore any points where the function jumps around, provided that such points are isolated. A slightly jittery function is one thing; a non-measurable set is a very different animal. Imagine a three-dimensional shape so fantastically intricate, so jagged and crinkled, that it is impossible to measure its volume and this gives some idea of the concept of non-measurability. From it flow such bizarre conclusions as the Banach-Tarski paradox.