# Prerequisites Measure theory for ug student in physics

• Analysis
• azeow
In summary: It is recommended to have a strong foundation in real analysis, and depending on how your university teaches the topic, you may need to take an analysis class to prepare. Additionally, some sources recommend having a background in functional analysis before studying measure theory.

#### azeow

Hi,

I would like to know if an undergraduate student in physics could be able to study measure theory in order to have a better understanding of the probability theory and go further in this way (stochastic process) ?
Assuming a first year of calculus and the level of "Mathematical methods in the physical sciences" by Mary L. Boas in mathematicsThank you a lot !

Yes. It could be covered in a theoretical calculus class or a real analysis class. It is more of a question of what level of sophistication your university teaches it. Check with your math department. Depending on how they present the topic, you might have to take an analysis class in order to prepare.

azeow
caz said:
Yes. It could be covered in a theoretical calculus class or a real analysis class. It is more of a question of what level of sophistication your university teaches it. Check with your math department. Depending on how they present the topic, you might have to take an analysis class in order to prepare.
Thank you a lot for your answer, my first message is a bit ambiguous and I apologize for that.

Indeed I would like to self study measure theory since I have no class on that, but I have a good reference (final goal I can say) for measure theory (and integration), there is the table of content below :

I Measure theory
1 Algebras and tribes of parts of a set
1.1 Definitions
1.2 Generated tribe, reciprocal image tribe
1.3 Examples
1.4 Product of measurable spaces
1.5 The Borelian tribe
2 Measure, measured space
2.1 Definitions
2.2 Elementary properties, characterisation of a finite measure
3 Extension of a measure and applications
3.1 Extension Theorem
3.2 External measure
3.3 Application: the Borel-Lebesgue measure
3.4 Application: Stieljes Measures on R
3.5 Finite product of a family of measured spaces
4 Complements: complete tribes, regularity of measures
4.1 Negligible sets, tribe and completed measure
4.2 Regularity of measures on a metric space
5 Measurable applications
5.1 Definition of a measurable application
5.2 General properties
5.3 Properties of real measurable functions
5.4 Function with value in R¯ = [-∞, +∞]
5.5 Transport of a measure, image measure
5.6 Approximation of a real measurable function
6 Measurement theory and probability
6.1 Introduction
6.2 Elementary Examples
6.2.1 Finite set: Ω = {ω1, ...ωn}
6.2.2 Case of an infinite enumerable set Ω = {ωi, i ∈ N}
6.3 Conditional probabilities, independent events
6.4 Random Variables
6.4.1 Real Random Variables
6.4.2 Random variables, random vectors, independence

II Integration
7 Integration of Positive Measurable Functions
7.1 (Higher) Integral of Stepped Functions
7.2 Integral of a positive measurable function
7.3 Property true almost everywhere
7.4 General Properties
7.5 Transfer theorem (change of variable)
7.6 Measures defined by densities
7.7 Absolute continuous, foreign measures
7.8 Absolute continuity and density
7.9 Change of variable theorem, λ Lebesgue measure
7.10 Characterisation of the product measure, Fubini-Tonelli theorem
8 Integration of any measurable functions
8.1 Integrale of a measurable function
8.1.1 Definitions
8.1.2 The set L1
8.2 General Properties
8.2.1 First properties, Fatou's lemma
8.2.2 Theorem of domin'ee convergence and applications
8.2.3 Examples
8.3 Fubini's Theorem for any measurable functions
8.3.1 Fubini's Theorem
8.3.2 Examples
8.4 Convolution
8.4.1 Convolution of Two Measures
8.4.2 Convolution of a Function and a Measure, of Two Functions
8.4.3 Examples
9 Integration Theory and Probability
9.1 Expectation and Moments
9.1.1 Expectation
9.1.2 Moments
9.1.3 Covariance and correlation
9.1.4 Properties of moments
9.1.5 Inequalities
9.2 Real random variable (random vector) and density
9.3 Return on independence

III Functional Analysis
10 The spaces Lp and Lp, p ∈ N* ∪ +{∞}
10.1 Definitions of Lp spaces
10.1.1 The spaces Lp, p ∈ N*
10.1.2 The spaces L∞, L∞
10.2 Properties of Lp spaces, 1 ≤ p ≤ +∞
10.2.1 ||.||p is a norm
10.2.2 Completeness of Lp spaces
10.2.3 Other Properties
10.3 Dual of Lp-spaces
10.4 Some results of functional analysis in L1(R, BR, λ)
11 The Fourier transform
11.1 Definitions
11.2 General properties
11.3 Examples
11.4 General properties X = Rd: injectivity and inversion theorems
11.4.1 Injectivity Theorem
11.4.2 Inversion Theorem
11.5 Analytical properties (on R)
11.6 Fourier transform in L1: analytical properties
11.7 Fourier transform in L2

Sorry if some notations are unusual this is from my reference's lecture notes which is in french

Last edited:
I briefly saw your deleted response. I believe the question you want to ask is

Is there a good reference to learn the theoretical foundations of probability (measure theory, lebesgue intergation, functional analysis, etc.) at the undergraduate level.

Other than saying you need to learn some real analysis, I cannot answer it.

azeow
Yes I don't know why but my response is not visible to everyone " This message is awaiting moderator approval, and is invisible to normal visitors."

But I see that you were able to see the outline from your answer thank you !
The table of content you saw (functional analysis, lebesgue integration and measure theory) corresponds to the content of my measure theory course. I don't seek about reference on that, just advices from people with a similar background than me who tried measure theory before, to know if my level in mathematics is sufficient to dive in. aha

Thank you a lot !

Unless it is more applied than I expect, I would try something like Abbott Understanding Analysis first. Theoretical math is different than the applied stuff.

azeow said:
Yes I don't know why but my response is not visible to everyone " This message is awaiting moderator approval, and is invisible to normal visitors."
The forum AI has a mind of its own. Often it is right, often it overreacts. Your reply has been restored; sorry for the delay.

caz said:
Unless it is more applied than I expect, I would try something like Abbott Understanding Analysis first. Theoretical math is different than the applied stuff.
This seems reasonable since I have Abbott's book so there is no cost to start with and the basic probability class that follows this course is heavy proof oriented and abstract, tank you !
berkeman said:
The forum AI has a mind of its own. Often it is right, often it overreacts. Your reply has been restored; sorry for the delay.
No problem ! Tank you !

berkeman
Pre-requisites to study measure theory?
... it's as much a maturity prerequisite as a prerequisite for actual concepts and techniques.
The actual prerequisites for abstract measure theory are basic set theory but you need some analysis to understand examples and counterexamples.

azeow
Keith_McClary said:
Pre-requisites to study measure theory?

The actual prerequisites for abstract measure theory are basic set theory but you need some analysis to understand examples and counterexamples.
Thank you a lot ! Your opinion confirms what Caz said

You need to understand how to read and write proofs in order to learn Measure Theory. I have a math BS and have not started it myself, but it is possible. (I have just been reviewing previous learned math, have not had time to do so).

Anyhow, Analysis is a must (you have Abbot which is a good introductory). Then you can jump into a Measure Theory Text, but the ones I have seen, require a more rigorous understanding of Analysis. Ie., Something like Pugh, Zorich, or Apostol.

For Functional Analysis. Theres a nice book by Simmons: Topology and Modern Analysis. First half is Topology, and second part is Functional Analysis. A nice read.

Theres also the book published by Matrix Editions titled Functional Analysis. It is no longer in print, but you can purchase a pdf etc. The only requirements for this one is Calculus. A good intro, but not as demanding as what you would probably encounter in a more rigorous text.

azeow
MidgetDwarf said:
You need to understand how to read and write proofs in order to learn Measure Theory. I have a math BS and have not started it myself, but it is possible. (I have just been reviewing previous learned math, have not had time to do so).

Anyhow, Analysis is a must (you have Abbot which is a good introductory). Then you can jump into a Measure Theory Text, but the ones I have seen, require a more rigorous understanding of Analysis. Ie., Something like Pugh, Zorich, or Apostol.

For Functional Analysis. Theres a nice book by Simmons: Topology and Modern Analysis. First half is Topology, and second part is Functional Analysis. A nice read.

Theres also the book published by Matrix Editions titled Functional Analysis. It is no longer in print, but you can purchase a pdf etc. The only requirements for this one is Calculus. A good intro, but not as demanding as what you would probably encounter in a more rigorous text.

The table of contents I put here is the most advanced version of a measure and integration theory course that can be taught (at least in France, in the third year) so I realize with your message that I have to acquire a better reflection than the one proposed through Mary L. Boas' book typically aha (which is a good lauching pad to more rigourous book in mathematics I have to say).The book provides by Abott is great from what I’ve seen until now and you are not the first to say that is a good introducy so I will pursue with I think !
I will take a look of Simmons book since I have never had a formal class in Topology it could be interesting

Thank you a lot !

## What is measure theory and why is it important for physics students?

Measure theory is a branch of mathematics that deals with the concept of measuring sets and their associated properties. It is important for physics students because it provides a rigorous framework for understanding and analyzing physical phenomena, especially in the field of quantum mechanics.

## What are the prerequisites for learning measure theory as an undergraduate physics student?

The prerequisites for learning measure theory as an undergraduate physics student typically include a strong foundation in calculus, linear algebra, and probability theory. It is also helpful to have some familiarity with real analysis and abstract algebra.

## How does measure theory relate to other areas of physics?

Measure theory has applications in various areas of physics such as statistical mechanics, quantum field theory, and general relativity. It provides a mathematical framework for understanding and analyzing physical systems and their properties.

## What are some common challenges that students face when learning measure theory?

Some common challenges that students face when learning measure theory include understanding the abstract concepts and notation, as well as applying the theory to solve problems in physics. It can also be challenging to grasp the connections between measure theory and other areas of mathematics and physics.

## How can students effectively learn measure theory for physics?

To effectively learn measure theory for physics, students should focus on understanding the fundamental concepts and definitions, practice solving problems, and make connections to other areas of physics. It is also helpful to seek out additional resources such as textbooks, online lectures, and study groups.