26.1. RANDOM VARIABLES AND INDEPENDENCE 717

Definition 26.1.7 A measure, µ defined on B (E) for E a separable metric spacewill be called inner regular if for all F ∈B (E) ,

µ (F) = sup{µ (K) : K ⊆ F and K is closed}

A measure, µ defined on B (E) will be called outer regular if for all F ∈B (E) ,

µ (F) = inf{µ (V ) : V ⊇ F and V is open}

When a measure is both inner and outer regular, it is called regular.

Note that if the metric space isRp then λX can be considered a Radon measure becauseyou can use it to obtain a positive linear functional and then use the Riesz representationtheorem for these.

For probability measures, the above definition of regularity tends to come free. Noteit is a little weaker than the usual definition of regularity because K is only assumed to beclosed, not compact. This is stated for convenience. It is Lemma 9.8.4 on Page 253.

Lemma 26.1.8 Let µ be a finite measure defined on B (E) where E is a metric space.Then µ is regular.

One can say more if the metric space is complete and separable. In fact in this case theabove definition of inner regularity can be shown to imply the usual one where the closedsets are replaced with compact sets. It is Lemma 9.8.5 on Page 255.

Lemma 26.1.9 Let µ be a finite measure on a σ algebra containing B (X) , the Borelsets of X , a separable complete metric space. (Polish space) Then if C is a closed set,

µ (C) = sup{µ (K) : K ⊆C and K is compact.}

It follows that for a finite measure on B (X) where X is a Polish space, µ is inner regularin the sense that for all F ∈B (X) ,

µ (F) = sup{µ (K) : K ⊆ F and K is compact}

Definition 26.1.10 A measurable functionX : (Ω,F ,µ)→ Z a topological spaceis called a random variable when µ (Ω) = 1. For such a random variable, one can definea distribution measure λX on the Borel sets of Z as follows:λX (G)≡ µ

(X−1 (G)

). This

is a well defined measure on the Borel sets of Z because it makes sense for every G openand G ≡

{G⊆ Z :X−1 (G) ∈F

}is a σ algebra which contains the open sets, hence the

Borel sets. Such a measurable function is also called a random vector.

Corollary 26.1.11 LetX be a random variable (random vector) with values in a com-plete metric space, Z. Then λX is an inner and outer regular measure defined on B (Z).

Proposition 26.1.12 For X a random vector defined above, X having values in acomplete separable metric space Z, then λX is inner and outer regular and Borel.

(Ω,P) X→ (Z,λX)h→ E

If h is Borel measurable and h ∈ L1 (Z,λX ;E) for E a Banach space, then∫Ω

h(X (ω))dP =∫

Zh(x)dλX . (26.1)