Wiener measure in a space of functions of two variables
J. Yeh · Transactions of the American Mathematical Society · 1960
The Wiener space C2 of functions of two variables is the collection of continuous functions }x(/, t) } on the unit square OSt, tSI satisfying x(0, t) =x(<, 0) =0.Integration on this space was first introduced by T. Kitagawa in his Analysis of variance applied to function spaces [2].There he defined the Wiener integral for functionals on C2 of the type Hixiti, T-f), • • • , x(/r, rf)) where i?(r?n, • • • , ??") is a function of rs real variables {rjhk} ih=l,2, ■ ■ • ,r,k = l,2, -• • , s) and {th}, {r*} are preassigned division points of the unit intervals O^fgl, O^r^l satisfying 0 = /0^i S ■ • • StrStr+i = l,0=ToSTiS • ■ ■ St8St,+i = 1.The Wiener integral for this class of functionals was defined to be p w I Hixiti, n), • • • , *(/,, T,))dwx /x /» oo r B ■ • • I P(l?ll, ' -• , Irs) II II piAh,k)drni ■ • • dt]r,s