# Measure Theory/Basic Structures And Definitions/Measures

In this section, we study measure spaces and measures.

## Contents

## Measure Spaces[edit]

Let be a set and be a collection of subsets of such that is a σ-ring.

We call the pair a **measure space**. Members of are called **measurable sets**.

A positive real valued function defined on is said to be a **measure** if and only if,

(i) and

(i)"Countable additivity": , where are pairwise disjoint sets.

we call the triplet a **measurable space**

A **probability measure** is a measure with total measure one (i.e., μ(*X*)=1); a probability space is a measure space with a probability measure.

## Properties[edit]

Several further properties can be derived from the definition of a countably additive measure.

### Monotonicity[edit]

is monotonic: If and are measurable sets with then .

### Measures of infinite unions of measurable sets[edit]

is subadditive: If , , , ... is a countable sequence of sets in , not necessarily disjoint, then

- .

is continuous from below: If , , , ... are measurable sets and is a subset of for all *n*, then the union of the sets is measurable, and

- .

### Measures of infinite intersections of measurable sets[edit]

is continuous from above: If , , , ... are measurable sets and is a subset of for all *n*, then the intersection of the sets is measurable; furthermore, if at least one of the has finite measure, then

- .

This property is false without the assumption that at least one of the has finite measure. For instance, for each *n* ∈ **N**, let

which all have infinite measure, but the intersection is empty.

## Examples[edit]

## Counting Measure[edit]

Start with a set Ω and consider the sigma algebra X on Ω consisting of all subsets of Ω. Define a measure μ on this sigma algebra by setting μ(A) = |A| if A is a finite subset of Ω and μ(A) = ∞ if A is an infinite subset of Ω, where |A| denotes the cardinality of set A. Then (Ω, X, μ) is a measure space. μ is called the counting measure.

## Lebesgue Measure[edit]

For *any* subset *B* of **R**^{n}, we can define an outer measure by:

- , and is a countable union of products of intervals .

Here, vol(*M*) is sum of the product of the lengths of the involved intervals. We then define the set *A* to be Lebesgue measurable if

for all sets *B*. These Lebesgue measurable sets form a σ-algebra, and the Lebesgue measure is defined by λ(*A*) = λ^{*}(*A*) for any Lebesgue measurable set *A*.