Does a completely regular space imply the Dirac measure. From wikipedia we have the definition:(adsbygoogle = window.adsbygoogle || []).push({});

Xis acompletely regular spaceif given any closed setFand any pointxthat does not belong toF, then there is a continuous function,f,fromXto the real lineRsuch thatf(x) is 0 and, for everyyinF,f(y) is 1.

And the Dirac measure is defined by:

A Dirac measure is a measure on a setXdefined for a givenx∈Xand any setA⊆Xby δ_{x}(A) = 0 for x∉A, and δ_{x}(A) = 1 for x∈A.

It seems the definition for a completely regular space includes the definition of a Dirac measure. The difference seems to be that the Dirac measure does not involve a continous function, but it does seem as though δ_{x}(A) = f(x), where the set A for the Dirac measure seems to be the same thing as the set F in the completely regular space. Both f(x)=δ_{x}(A)=0 if x∉F or x∉A and 1 otherwise.

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# Completely regular space and the Dirac measure

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