Tightness of measures

In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not “escape to infinity“.

Concept in measure theory
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. . . Tightness of measures . . .

Let

(X,T){displaystyle (X,T)}

be a Hausdorff space, and let

Σ{displaystyle Sigma }

be a σ-algebra on

X{displaystyle X}

that contains the topology

T{displaystyle T}

. (Thus, every open subset of

X{displaystyle X}

is a measurable set and

Σ{displaystyle Sigma }

is at least as fine as the Borel σ-algebra on

X{displaystyle X}

.) Let

M{displaystyle M}

be a collection of (possibly signed or complex) measures defined on

Σ{displaystyle Sigma }

. The collection

M{displaystyle M}

is called tight (or sometimes uniformly tight) if, for any

ε>0{displaystyle varepsilon >0}

, there is a compact subset

Kε{displaystyle K_{varepsilon }}

of

X{displaystyle X}

such that, for all measures

μM{displaystyle mu in M}

,

|μ|(XKε)ε.{displaystyle |mu |(Xsetminus K_{varepsilon })

where

|μ|{displaystyle |mu |}

is the total variation measure of

μ{displaystyle mu }

. Very often, the measures in question are probability measures, so the last part can be written as

μ(Kε)>1ε.{displaystyle mu (K_{varepsilon })>1-varepsilon .,}

If a tight collection

M{displaystyle M}

consists of a single measure

μ{displaystyle mu }

, then (depending upon the author)

μ{displaystyle mu }

may either be said to be a tight measure or to be an inner regular measure.

If

Y{displaystyle Y}

is an

X{displaystyle X}

-valued random variable whose probability distribution on

X{displaystyle X}

is a tight measure then

Y{displaystyle Y}

is said to be a separable random variable or a Radon random variable.

. . . Tightness of measures . . .

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. . . Tightness of measures . . .

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