2036 CHAPTER 61. PROBABILITY IN INFINITE DIMENSIONS

many of the ek the following condition holds. For every ε > 0 there exists Pε ∈F ({en})such that if P ∈F ({en}) and PPε = 0, then

ν ({x ∈ H : ||Px||> ε})< ε.

Then ||·|| is called Gross measurable.

The following lemma is a fundamental result about Gross measurable norms. It is aboutthe continuity of ||·|| . It is obvious that with respect to the topology determined by ||·|| thatx→ ||x|| is continuous. However, it would be interesting if this were the case with respectto the topology determined by the norm on H, |·| . This lemma shows this is the case andso the funny condition above implies x→ ||x|| is a continuous, hence Borel measurablefunction.

Lemma 61.9.10 Let ||·|| be Gross measurable. Then there exists c > 0 such that

||x|| ≤ c |x|

for all x ∈ H. Furthermore, the above definition is well defined.

Proof: First it is important to consider the question whether the above definition is welldefined. To do this note that on PH, the two norms are equivalent because PH is a finitedimensional space. Let G = {y ∈ PH : ||y||> ε} so G is an open set in PH. Then

{x ∈ H : ||Px||> ε}

equals{x ∈ H : Px ∈ G}

which equals a set of the form{x ∈ H :

((x,ei1)H , · · · ,(x,eim)H

)∈ G′

}for G′ an open set in Rm and so everything makes sense in the above definition.

Now it is necessary to verify ||·|| ≤ c |·|. If it is not so, there exists e1 such that

||e1|| ≥ 1, |e1|= 1.

Suppose {ek}nk=1 have been chosen such that each is a unit vector in H and ||ek|| ≥ k. Then

considering span(e1, · · · ,en)⊥ if for every x ∈ span(e1, · · · ,en)

⊥ , ||x|| ≤ c |x| , then if z ∈His arbitrary, z = x+y where y ∈ span(e1, · · · ,en) and so since the two norms are equivalenton a finite dimensional subspace, there exists c′ corresponding to span(e1, · · · ,en) such that

||z||2 ≤ (||x||+ ||y||)2 ≤ 2 ||x||2 +2 ||y||2

≤ 2c2 |x|2 +2c′ |y|2

≤(2c2 +2c′2

)(|x|2 + |y|2

)=

(2c2 +2c′2

)|z|2