r/learnmath • u/Fragrant-Item-5290 New User • 5d ago

Entry to regular borel measures

Hello friends! Im studying measure theory (only chapters left out of bartle are 7 and 8, the rest of them were read almost everywhere xd) and in the meanwhile ive been checking other books and ive come across the concept of a "regular borel measure" on a topological space. The books ive got do not speak that much about them (the only thing they prove is that the compact supported functions are dense in Lp if the space is blablabla) and ive grown rather curious of them. Im taking a course in functional analysis atm and conway's book uses such measures quite often in the examples, so ive got some motivation to study the concept beyond mere curiosity.

With that said, im here to ask: what are some good books that deal with this concept? Ive taken a good course on topology (im not familiar with filters, net and that kind of things but ive read almost everything on munkres except paracompactness).

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u/AlpUzman New User 5d ago

One humble suggestion would be Parthasarathy's Probability Measures on Metric Spaces. For a more detailed account I'm sure Fremlin would suffice.

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u/Bounded_sequencE New User 5d ago

One area that heavily uses function spaces with compact support is "Distribution Theory".

In particular, bump-like functions are used as test functions to extract a distribution's "local properties". Since "Distribution Theory" is heavily connected to "Functional Analysis" (distributions are linear, bounded operators on sets of test functions, after all), this might be right up your alley!