Strictly positive measure

In mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure is one that is "nowhere zero", or that is zero "only on points".

Definition

Let

(

X

,

T

)

{\displaystyle (X,T)}

be a Hausdorff topological space and let

Σ

{\displaystyle \Sigma }

be a

σ

{\displaystyle \sigma }

-algebra on

X

{\displaystyle X}

that contains the topology

T

{\displaystyle T}

(so that every open set is a measurable set, and

Σ

{\displaystyle \Sigma }

is at least as fine as the Borel

σ

{\displaystyle \sigma }

-algebra on

X

{\displaystyle X}

). Then a measure

μ

{\displaystyle \mu }

on

(

X

,

Σ

)

{\displaystyle (X,\Sigma )}

is called strictly positive if every non-empty open subset of

X

{\displaystyle X}

has strictly positive measure.

More concisely,

μ

{\displaystyle \mu }

is strictly positive if and only if for all

U

∈

T

{\displaystyle U\in T}

such that

U

≠

∅

,

μ

(

U

)

>

0.

{\displaystyle U\neq \varnothing ,\mu (U)>0.}

Examples

Counting measure on any set

X

{\displaystyle X}

(with any topology) is strictly positive.

Dirac measure is usually not strictly positive unless the topology

T

{\displaystyle T}

is particularly "coarse" (contains "few" sets). For example,

δ

0

{\displaystyle \delta _{0}}

on the real line

R

{\displaystyle \mathbb {R} }

with its usual Borel topology and

σ

{\displaystyle \sigma }

-algebra is not strictly positive; however, if

R

{\displaystyle \mathbb {R} }

is equipped with the trivial topology

T

=

{

∅

,

R

}

,

{\displaystyle T=\{\varnothing ,\mathbb {R} \},}

then

δ

0

{\displaystyle \delta _{0}}

is strictly positive. This example illustrates the importance of the topology in determining strict positivity.

Gaussian measure on Euclidean space

R

n

{\displaystyle \mathbb {R} ^{n}}

(with its Borel topology and

σ

{\displaystyle \sigma }

-algebra) is strictly positive.