Re: When the Gilbert space with measure is separable

"Kostyantyn Yusenko" <kay.math@xxxxxxxxx> writes:

We are given a Gilbert space H and measure m. The question is for

[ Hilbert space, of course ]

which measures m the spaces L_2(H,m) are separable?

Is this true that when m is propabilistic measure then space L_2(H,m)
is separable?

I'm not sure what you mean by L_2(H,m). I can think of two plausible
1) m is a measure on H, and you're dealing with scalar-valued functions
f on H where ||f||^2 = int_H |f(x)|^2 dm(x) < infty
2) m is a measure on some measurable space X, and you're dealing with
H-valued functions f on X where ||f||^2 = int_X ||f(x)||^2 dm(x) < infty.

In either case, you certainly want H to be separable. Whether m is
a probability measure is not really relevant, but you do want it to be
sigma-finite.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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