59.1. RANDOM VARIABLES AND INDEPENDENCE 1857

If X is strongly measurable, then it is weakly measurable and so each fi ◦X is a real (com-plex) valued measurable function. Hence the expression on the right in the above is mea-surable. Now if U is any open set in Z, then it is the countable union of such closed disksU = ∪iDi. Therefore, X−1 (U) = ∩iX−1 (Di) ∈ S . It follows that strongly measurableimplies inverse images of open sets are in S .

Conversely, suppose X−1 (U) ∈ S for every open U . Then for f ∈ Z′, f ◦X is realvalued and measurable. Therefore, X is weakly measurable. By the Pettis theorem, itfollows that f ◦X is strongly measurable.

Proposition 59.1.6 If X : Ω→ Z is measurable, then σ (X) equals the smallest σ algebrasuch that X is measurable with respect to it. Also if Xi are random variables having valuesin separable Banach spaces Zi, then σ (X) = σ (X1, · · · ,Xn) where X is the vector mappingΩ to ∏

ni=1 Zi and σ (X1, · · · ,Xn) is the smallest σ algebra such that each Xi is measurable

with respect to it.

Proof: Let G denote the smallest σ algebra such that X is measurable with respectto this σ algebra. By definition X−1 (open) ∈ G . Furthermore, the set of all E such thatX−1 (E) ∈ G is a σ algebra. Hence it includes all the Borel sets. Hence X−1 (Borel) ∈ Gand so G ⊇ σ (X) . However, σ (X) defined above is a σ algebra such that X is measurablewith respect to σ (X) . Therefore, G = σ (X).

Letting Bi be a Borel set in Zi,∏ni=1 Bi is a Borel set by Proposition 59.1.3 and so

X−1

(n

∏i=1

Bi

)= ∩n

i=1X−1i (Bi) ∈ σ (X1, · · · ,Xn)

If G denotes the Borel sets F ⊆ ∏ni=1 Zi such that X−1 (F) ∈ σ (X1, · · · ,Xn) , then G is

clearly a σ algebra which contains the open sets. Hence G = B the Borel sets of ∏ni=1 Zi.

This shows that σ (X)⊆ σ (X1, · · · ,Xn) . Next we observe that σ (X) is a σ algebra with theproperty that each Xi is measurable with respect to σ (X) . This follows from X−1

i (Bi) =

X−1(

∏nj=1 A j

)∈ σ (X) , where each A j = Z j except for Ai = Bi.Since σ (X1, · · · ,Xn) is

defined as the smallest such σ algebra, it follows that σ (X)⊇ σ (X1, · · · ,Xn) .For random variables having values in a separable Banach space or even more generally

for a separable metric space, much can be said about regularity of λ X.

Definition 59.1.7 A measure, µ defined on B (E) will be called inner regular if for allF ∈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.

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.