Reference request - Measure theory

In summary, there are a few recommended books for an introductory understanding of measure theory, including "Lebesgue Integration on Euclidean Space" by Frank Jones, "Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics" by A. Lasota and M.C. Mackey, "A User-Friendly Introduction to Lebesgue Measure and Integration" by Gail S. Nelson, and "The Lebesgue Integral for Undergraduates" by William Johnston. These books provide a solid foundation for understanding measure theory and its application in ergodic theory and probability theory.
  • #1
Joppy
MHB
284
22
Hi!

Can anyone recommend a good introductory book for measure theory? I've found Terence Tao's online book to be a good start, but would I be asking too much if I wanted something even more introductory?

Ultimately I'm working toward Ergodic theory (and probability theory along the way) with an applied focus if that helps.

Thanks.
 
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  • #2
Joppy said:
Hi!

Can anyone recommend a good introductory book for measure theory? I've found Terence Tao's online book to be a good start, but would I be asking too much if I wanted something even more introductory?

Ultimately I'm working toward Ergodic theory (and probability theory along the way) with an applied focus if that helps.

Thanks.
You may find Chapters 2-5 of the following book relevant and helpful ...

"Lebesgue Integration on Euclidean Space" by Frank Jones (Revised Edition) Jones and Bartlett Publishers, 2001

Peter
 
  • #3
Peter said:
You may find Chapters 2-5 of the following book relevant and helpful ...

"Lebesgue Integration on Euclidean Space" by Frank Jones (Revised Edition) Jones and Bartlett Publishers, 2001

Peter

Thanks Peter! Having a look now.
 
  • #4
Joppy said:
Ultimately I'm working toward Ergodic theory (and probability theory along the way) with an applied focus if that helps.

A somewhat unusual book recommended to me once by one of my bosses is Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics by A. Lasota and M.C. Mackey. I am usually a bit allergic for terms such as "chaos" and "fractals", but in this case they should not be deterrents.

The book is not about measure theory itself, but rather it develops those aspects of measure theory that are needed to understand the more "applied" stochastic aspects of various classes of deterministic dynamical systems. It does so in a rigorous but very accessible way, assuming just a good foundation in multivariable calculus. I enjoyed it a lot.
 
  • #5
Krylov said:
I am usually a bit allergic for terms such as "chaos" and "fractals", but in this case they should not be deterrents.

That's the good stuff isn't it! :p.
Krylov said:
The book is not about measure theory itself, but rather it develops those aspects of measure theory that are needed to understand the more "applied" stochastic aspects of various classes of deterministic dynamical systems. It does so in a rigorous but very accessible way, assuming just a good foundation in multivariable calculus. I enjoyed it a lot.

Sounds great, just the sort of thing I was thinking of.
 
  • #6
Peter said:
You may find Chapters 2-5 of the following book relevant and helpful ...

"Lebesgue Integration on Euclidean Space" by Frank Jones (Revised Edition) Jones and Bartlett Publishers, 2001

Peter

Honestly I think this is the first mathematical text I've read were I feel as though the author really makes an attempt to explain the reasoning and motivation. Thanks again.
 
  • #7
Joppy said:
Honestly I think this is the first mathematical text I've read were I feel as though the author really makes an attempt to explain the reasoning and motivation. Thanks again.
Joppy,

I have just come across another possibility for you ...

"A User-Friendly Introduction to Lebesgue Measure and Integration" by Gail S. Nelson (AMS 2015)
Yet another book, but more on the Lebesgue integral than Lebesgue measure (although chapter 4 is on measure theory) is the following:

"The Lebesgue Integral for Undergraduates" by William Johnston (MAA Press, 2015)Hope that helps in some way ...

Peter
 
  • #8
Peter said:
Joppy,

I have just come across another possibility for you ...

"A User-Friendly Introduction to Lebesgue Measure and Integration" by Gail S. Nelson (AMS 2015)
Yet another book, but more on the Lebesgue integral than Lebesgue measure (although chapter 4 is on measure theory) is the following:

"The Lebesgue Integral for Undergraduates" by William Johnston (MAA Press, 2015)Hope that helps in some way ...

Peter

Great! Thanks a lot.
 

FAQ: Reference request - Measure theory

What is measure theory?

Measure theory is a branch of mathematics that deals with the concept of "measure", which is a mathematical way of quantifying the size or extent of a set. It provides a rigorous framework for understanding and manipulating the concept of size, and has applications in various fields such as probability, analysis, and geometry.

Why is measure theory important?

Measure theory is important because it provides a rigorous and systematic way of defining and manipulating the concept of size, which is fundamental in many areas of mathematics and science. It allows for precise and consistent calculations and proofs, and has applications in fields such as probability, statistics, and physics.

What are the main concepts in measure theory?

The main concepts in measure theory are measures, measurable sets, and measurable functions. A measure is a function that assigns a non-negative number to a set, representing its size. Measurable sets are subsets of a given set that can be assigned a measure, and measurable functions are functions that preserve the measure of sets.

What are some applications of measure theory?

Measure theory has many applications in mathematics, physics, and other fields. In probability theory, it is used to define and calculate probabilities of events. In analysis, it is used to define and study integrals and derivatives. In geometry, it is used to define and measure geometric objects such as length, area, and volume.

Are there any prerequisites for learning measure theory?

It is recommended to have a solid understanding of basic mathematics, including calculus, set theory, and real analysis, before diving into measure theory. Familiarity with concepts such as limits, continuity, and convergence is also helpful. It is also important to have a strong understanding of mathematical proof techniques.

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