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

[mathmari]

Hey!

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)

 

[Math Amateur]

Hey!

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