In layman’s terms, “measures” are functions that are intended to represent ideas of length, area, mass, etc. The inputs for the measure functions would be sets, and the output would be a real value, possibly including infinity.
It would be desirable to attach the value 0 to the empty set and measures should be additive over disjoint sets in X.
Definition (from Bartle): A measure is an extended real-valued function defined on a -algebra X of subsets of X such that
(ii) for all
(iii) is countably additive in the sense that if is any disjoint sequence () of sets in X, then
If a measure does not take on , we say it is finite. More generally, if there exists a sequence of sets in X with and such that for all n, then we say that is -finite. We see that if a measure is finite implies it is -finite, but not necessarily the other way around.
Examples of measures
(a) Let X be any nonempty set and let X be the -algebra of all subsets of X. Let be definied on X by , for all . We can see that is finite and thus also -finite.
Let be defined by , if . is an example of a measure that is neither finite nor -finite.
The most famous measure is definitely the Lebesgue measure. If X=R, and X=B, the Borel algebra, then (shown in Bartle’s Chapter 9) there exists a unique measure defined on B which coincides with length on open intervals. I.e. if E is the nonempty interval (a,b), then . This measure is usually called Lebesgue measure (or sometimes Borel measure). It is not a finite measure since . But it is -finite since any interval can be broken down into a sequence of sets () such that for all n.