61.7. GAUSSIAN MEASURES 2015

61.7.2 Fernique’s TheoremThe following is an interesting lemma.

Lemma 61.7.4 Suppose µ is a symmetric Gaussian measure on the real separable Ba-nach space, E. Then there exists a probability space, (Ω,F ,P) and independent randomvariables, X and Y mapping Ω to E such that L (X) = L (Y ) = µ. Also, the two randomvariables,

1√2(X−Y ) ,

1√2(X +Y )

are independent and

L

(1√2(X−Y )

)= L

(1√2(X +Y )

)= µ.

Proof: Letting X ′ ≡ 1√2(X +Y ) and Y ′ ≡ 1√

2(X−Y ) , it follows from Theorem 59.13.2

on Page 1898, that X ′ and Y ′ are independent if whenever h1, · · · ,hm ∈ E ′ and g1, · · · ,gk ∈E ′, the two random vectors,(

h1 ◦X ′, · · · ,hm ◦X ′)

and(g1 ◦Y ′, · · · ,gk ◦Y ′

)are independent. Now consider linear combinations

m

∑j=1

t jh j ◦X ′+k

∑i=1

sigi ◦Y ′.

This equals

1√2

m

∑j=1

t jh j (X)+1√2

m

∑j=1

t jh j (Y )

+1√2

k

∑i=1

sigi (X)− 1√2

k

∑i=1

sigi (Y )

=1√2

(m

∑j=1

t jh j +k

∑i=1

sigi

)(X)

+1√2

(m

∑j=1

t jh j−k

∑i=1

sigi

)(Y )

and this is the sum of two independent normally distributed random variables so it is alsonormally distributed. Therefore, by Theorem 61.6.4(

h1 ◦X ′, · · · ,hm ◦X ′,g1 ◦Y ′, · · · ,gk ◦Y ′)

is a random variable with multivariate normal distribution and by Theorem 61.6.9 the tworandom vectors (

h1 ◦X ′, · · · ,hm ◦X ′)

and(g1 ◦Y ′, · · · ,gk ◦Y ′

)