MHB Books to Learn Measure Theory Theory: Borel, Lebesgue, Cantor Set & More

AI Thread Summary
Recommended books for learning measure theory include "Lebesgue Integration on Euclidean Space" by Frank Jones, which provides a basic introduction to Lebesgue integration and covers many relevant topics. Another suggestion is "The Lebesgue-Stieltjes Integral" by Michael Carter and Bruce van Brunt, focusing on the practical application of Lebesgue integrals. However, both texts emphasize integration over measure theory itself, which may not fully meet the request for a comprehensive measure theory resource. For a deeper understanding of measure theory concepts like Borel sets, Lebesgue measure properties, and convergence theorems, additional graduate-level texts may be necessary. These recommendations serve as a starting point for exploring the complexities of measure theory.
mathmari
Gold Member
MHB
Messages
4,984
Reaction score
7
Hey! :o

What book would you recommend me to read about measure theory and especially the following:

Measure and outer meansure, Borel sets, the outer Lebesgue measure.
The Cantor set.
Properties of Lebesgue measure (translation invariance, completeness, regularity, uniqueness).
Steinhaus theorem, non-Lebesgue measurable sets.
Measurable functions, integrable functions, convergence theorems.
Elementary theory of Hilbert spaces.
Complex measures, the Radon-Nikodym theorem.
The maximal function Hardy-Littlewood.
Differentiation of measures and functions.
Product of measures. The Fubini theorem.
Change of variable. Polar coordinates. Convolutions.

?? (Wondering)
 
Mathematics news on Phys.org
mathmari said:
Hey! :o

What book would you recommend me to read about measure theory and especially the following:

Measure and outer meansure, Borel sets, the outer Lebesgue measure.
The Cantor set.
Properties of Lebesgue measure (translation invariance, completeness, regularity, uniqueness).
Steinhaus theorem, non-Lebesgue measurable sets.
Measurable functions, integrable functions, convergence theorems.
Elementary theory of Hilbert spaces.
Complex measures, the Radon-Nikodym theorem.
The maximal function Hardy-Littlewood.
Differentiation of measures and functions.
Product of measures. The Fubini theorem.
Change of variable. Polar coordinates. Convolutions.

?? (Wondering)
Hello mathmari,

A book which gives a basic introduction to Lebesgue Integration and seem to cover most of your list is as follows:

"Lebesgue Integration on Euclidean space" by Frank Jones (Jones and Bartlett Publishers)

Another book which focuses on giving students the knowledge and skills to use the Lebesgue or Lebesgue-Stieltjes integrals is as follows:

"The Lebesgue-Stieltjes Integral" by Michael Carter and Bruce van Brunt (Springer)

Hope that helps ... ...If you are looking for a high level of generality and also rigour then possibly someone else can help with some more graduate level texts, but the books I have recommended will give you a gentle introduction to measure theory and Lebesgue integration although their emphasis is less on measure theory and more on integration ... ... so maybe I really have not answered your question ...

Best Regards,

Peter***EDIT***

Sorry mathmari,

I may have answered you request too quickly without studying your request ... ... as I have noted above I am recommending books that focus on Lebesgue Integration rather than just focussing on measure theory ... indeed the second book I mentioned is very focussed on integration and has very little on measure theory ...

Peter
 
Last edited:
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top