The idea behind (Lebesgue-) measures is to extend the notion of lengths for a larger class of subsets of than just finite intervals. This is particularly useful for integration as it forms the basis for Lebesgue Integration which allows one to integrate many more functions than Riemann integration as all Riemann-integrable functions are Lebesgue-integrable (as long as the integral is proper). Also, much of probability is based on measure theory. For further help in starting a wiki page, check out Wiki Guidelines and Wiki Formatting or come chat with us.
is called a -algebra on a set (generally referred to as the sample space) if it satisfies the following 3 properties:
The first two properties define an algebra on , whilst property (iii) adds that sets in this algebra must also be closed under countable union. This leads naturally to the definition of a measure space:
A pair where is a -algebra on is called a measurable space and any set in is called a measurable subset of with respect to
A function : is called a measure on if
(b) with disjoint
where condition (b) says that a measure must be countably additive (over disjoint sets). Also, note that the codomain is the extended real number line [0, ] rather than [0, ) such that is permitted.
Consider the triple . If is a measurable space and is a measure with then we have a probability space where is the sample space and is a set of subsets of containing events.
Let be the length of an interval For example,
The (Lebesgue) outer measure of any set is given by where for intervals which (we say) form a covering of
if and only if is a null set. One example of a null set is a singleton for . Now suppose we have a uniform distribution on Then as a single element forms a null set, we have that which comes as a bit of a shock as the event of picking is not impossible, yet the probability of it occurring is zero. More surprisingly despite the fact that there are infinitely many rationals in This is because there are countably many rationals and infinitely many reals (on any non-empty interval) so the set of rationals is 'negligible' compared to the set of reals. Thus, if we say that " almost surely won't occur."
Show that if thenWe will show that and then use the definition of the infimum of a set. Let be a sequence of covering intervals for then so is also a covering sequence of intervals for i.e. so . Taking the same we have that as the infimum is a lower bound for a set, so as was taken from is a lower for . Hence, as the infimum is the greatest lower bound (of a set), .