The Space of Signed Measures
The sum of two finite signed measures is a finite signed measure, as is the product of a finite signed measure by a real number: they are closed under linear combination. It follows that the set of finite signed measures on a measure space (X, Σ) is a real vector space; this is in contrast to positive measures, which are only closed under conical combination, and thus form a convex cone but not a vector space. Furthermore, the total variation defines a norm in respect to which the space of finite signed measures becomes a Banach space. This space has even more structure, in that it can be shown to be a Dedekind complete Banach lattice and in so doing the Radon–Nikodym theorem can be shown to be a special case of the Freudenthal spectral theorem.
If X is a compact separable space, then the space of finite signed Baire measures is the dual of the real Banach space of all continuous real-valued functions on X, by the Riesz representation theorem.
Read more about this topic: Signed Measure
Famous quotes containing the words space, signed and/or measures:
“What a phenomenon it has beenscience fiction, space fictionexploding out of nowhere, unexpectedly of course, as always happens when the human mind is being forced to expand; this time starwards, galaxy-wise, and who knows where next.”
—Doris Lessing (b. 1919)
“In 1869 he started his work for temperance instigated by three drunken men who came to his home with a paper signed by a saloonkeeper and his patrons on which was written For Gods sake organize a temperance society.”
—Federal Writers Project Of The Wor, U.S. public relief program (1935-1943)
“There are other measures of self-respect for a man, than the number of clean shirts he puts on every day.”
—Ralph Waldo Emerson (18031882)