Measure (mathematics)

If E₁ and E₂ are measurable sets with E₁ ⊆ E₂ then

${\displaystyle \mu (E_{1})\leq \mu (E_{2}).}$

For any countable sequence E₁, E₂, E₃, ... of (not necessarily disjoint) measurable sets E_n in Σ:

${\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)\leq \sum _{i=1}^{\infty }\mu (E_{i}).}$

If E₁, E₂, E₃, ... are measurable sets and ${\displaystyle E_{n}\subseteq E_{n+1},}$ for all n, then the union of the sets E_n is measurable, and

${\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)=\lim _{i\to \infty }\mu (E_{i})=\sup _{i\geq 1}\mu (E_{i}).}$

If E₁, E₂, E₃, ... are measurable sets and, for all n, ${\displaystyle E_{n+1}\subseteq E_{n},}$ then the intersection of the sets E_n is measurable; furthermore, if at least one of the E_n has finite measure, then

${\displaystyle \mu \left(\bigcap _{i=1}^{\infty }E_{i}\right)=\lim _{i\to \infty }\mu (E_{i})=\inf _{i\geq 1}\mu (E_{i}).}$

This property is false without the assumption that at least one of the E_n has finite measure. For instance, for each n ∈ N, let E_n = [n, ∞) ⊂ R, which all have infinite Lebesgue measure, but the intersection is empty.

A measurable set X is called a null set if μ(X) = 0. A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set is measurable.

A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set. One defines μ(Y) to equal μ(X).

If the ${\displaystyle \mu }$-measurable function ${\displaystyle f}$ takes values on ${\displaystyle [0,\infty ],}$ then

${\displaystyle \mu \{x:f(x)\geq t\}=\mu \{x:f(x)>t\},}$

for almost all ${\displaystyle t\in \mathbb {R} ,}$ with respect to the Lebesgue measure.[1] This property is used in connection with Lebesgue integral.

Measures are required to be countably additive. However, the condition can be strengthened as follows. For any set ${\displaystyle I}$ and any set of nonnegative ${\displaystyle r_{i},i\in I}$ define:

${\displaystyle \sum _{i\in I}r_{i}=\sup \left\lbrace \sum _{i\in J}r_{i}:|J|<\aleph _{0},J\subseteq I\right\rbrace .}$

That is, we define the sum of the ${\displaystyle r_{i}}$ to be the supremum of all the sums of finitely many of them.

A measure ${\displaystyle \mu }$ on ${\displaystyle \Sigma }$ is ${\displaystyle \kappa }$-additive if for any ${\displaystyle \lambda <\kappa }$ and any family of disjoint sets ${\displaystyle X_{\alpha },\alpha <\lambda }$ the following hold:

${\displaystyle \bigcup _{\alpha \in \lambda }X_{\alpha }\in \Sigma }$

${\displaystyle \mu \left(\bigcup _{\alpha \in \lambda }X_{\alpha }\right)=\sum _{\alpha \in \lambda }\mu \left(X_{\alpha }\right).}$

Note that the second condition is equivalent to the statement that the ideal of null sets is ${\displaystyle \kappa }$-complete.

A measure space (X, Σ, μ) is called finite if μ(X) is a finite real number (rather than ∞). Nonzero finite measures are analogous to probability measures in the sense that any finite measure μ is proportional to the probability measure ${\displaystyle {\frac {1}{\mu (X)}}\mu }$. A measure μ is called σ-finite if X can be decomposed into a countable union of measurable sets of finite measure. Analogously, a set in a measure space is said to have a σ-finite measure if it is a countable union of sets with finite measure.

For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals [k, k+1] for all integers k; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to the Lindelöf property of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.

A measure is said to be s-finite if it is a countable sum of bounded measures. S-finite measures are more general than sigma-finite ones and have applications in the theory of stochastic processes.