Talk:Projection-valued measure

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[edit] Standard Borel space

Is a standard Borel space the same as a Borel space? -- 10:51, 28 October 2005 Oleg Alexandrov

Standard Borel space: Borel structure of a Polish space. -- 19:12, 28 October 2005 CSTAR

Great! Can I challenge you to fill in the redlink? Oleg Alexandrov (talk) 09:57, 29 October 2005 (UTC)

[edit] Vector measure

Re: recent addition:

It is further generalized by vector measures which take values in any Banach space.

This is actually quite subtle. By definition, a projection-valued measure P is σ-additive relative to the weak (or strong) operator topology on the set of self-adjoint projections. This means the P measure of a monotone union of a sequence of sets is the operator weak limit of the P measures of the elements of the sequence. This is equivalent to P being σ-additive relative to the ultraweak topology, since the ultrweak and weak operator toplogies coincide on bounded sets. The ultraweak topology in turn is a weak* topology (the algebra L(H) of bounded operators on H is the dual of the Banach space of trace class operators). However, weak* topologies require additional structure to define, i.e. being a dual space. Note that the weak* topology on a dual Banach space E* is weaker in general than the weak topology on E* considered as a Banach space in its own right.

So one should actually say that a projection-valued measure is generalized by a vector measure which which takes values in a dual Banach space and is weak* continuous.

I hope I haven't said anything silly here. --CSTAR 17:03, 9 February 2007 (UTC)

I have a weak* headache after reading all that. But thanks, I got your point. Oleg Alexandrov (talk) 02:44, 10 February 2007 (UTC)