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

In summary, for a basic introduction to measure theory with a focus on Lebesgue integration, "Lebesgue Integration on Euclidean Space" by Frank Jones and "The Lebesgue-Stieltjes Integral" by Michael Carter and Bruce van Brunt are recommended. However, for a more advanced and rigorous approach, other graduate level texts may be more suitable.
  • #1
mathmari
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Hey! :eek:

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)
 
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  • #2
mathmari said:
Hey! :eek:

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

What is Measure Theory?

Measure Theory is a branch of mathematics that deals with the study of measures on sets. It provides a rigorous and comprehensive framework for defining and analyzing the concept of "size" or "magnitude" of sets.

What are some important topics covered in Books on Measure Theory?

Some important topics covered in books on Measure Theory include Borel and Lebesgue measures, Cantor sets, integration theory, and probability measures. These topics are essential for understanding advanced topics in analysis, probability, and statistics.

How is Measure Theory used in real-world applications?

Measure Theory has a wide range of applications in various fields, including physics, economics, and engineering. It is used to analyze and model complex systems, such as traffic flow, fluid dynamics, and financial markets.

Are there any prerequisites for studying Measure Theory?

Yes, a strong background in real analysis, including topics such as limits, continuity, and differentiation, is essential for studying Measure Theory. Knowledge of basic set theory and mathematical proofs is also helpful.

How can I improve my understanding of Measure Theory?

Aside from reading books on Measure Theory, it is important to practice solving problems and proofs to strengthen your understanding of the concepts. Attending lectures, seminars, and workshops on the subject can also be beneficial. Additionally, discussing and collaborating with others can help clarify any misunderstandings or gaps in knowledge.

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