Measures

The concept of a measure is a generalization and formalization of geometric measures (length, area, volume), and other common notions such as mass and probability.

Let be a set and be a -algebra over . A set function is called a measure if and only if it satisfies the following properties:

1. Non-negativity: For all ,
2. Null empty set:
3. It is Sigma Additive

The pair is called a measurable space and the members of are called measurable sets.

The triple is called a measure space, and is called the measure.

A probability space is a measure space with a probability measure.

If and are measurable sets with (subset), then:

For any countable sequence of measurable sets , not necessarily disjoint, we have:

The distinction between subaddivitiy and additivity is that the sets might not be disjoint, and as such duplicates are discarded.

For all measurable sets , if the sets are increasing (i.e ), then the union of the sets is measurable, and

(see supremum)

As increases, each set includes either the same elements or more elements than the previous set. As such, we have the recurrence relation .

For all measurable sets , if the sets are decreasing (i.e ), then the intersection of of the sets is measurable, and, and

(see infimum)

This property is very similar to the previous one. The explanation is also nearly identical.
As increases, each set includes either the same elements or fewer elements than the previous set. As such, we have the recurrence relation .