# Basic measure theory for physics students

• Analysis
• lizzie96'
In summary, the conversation is about the need for knowledge on measure theory for understanding Quantum Theory. The recommended resources include a brief chapter on measure theory from Wikipedia, a short book by Bartle, a more geometrical book by Jones, and a book by Conway, which also covers functional analysis. The first few chapters from each of these resources are sufficient to gain enough knowledge of measure theory for understanding Quantum Theory. It is also mentioned that some people may not have any issues understanding Quantum Theory without prior knowledge of measure theory.
lizzie96'
I'm trying to read Brian Hall's book "Quantum Theory for Mathematicians". While (I think) I have a basic grasp of most of the prerequisites, I don't know any measure theory. According to the appendix, presumed knowledge includes "the basic notions of measure
theory, including the concepts of σ-algebras, measures, measurable functions, and the Lebesgue integral". Could anyone recommend a short book/ online notes that give me just enough knowledge of measure theory for QM?

This book has a brief chapter on measure theory.

A quite short and good book is Bartle: https://www.amazon.com/dp/0471042226/?tag=pfamazon01-20 You only need to read the first 6 chapters, the other chapters are nice, but not as important for your goal.

A very nice and more geometrical book is Jones, but this is longer than Bartle, so it would take more time: https://www.amazon.com/dp/0763717088/?tag=pfamazon01-20

If you're into functional analysis (like your post suggests), you could try the book by Conway: The first four chapters are enough, and it will additionally do some things with Hilbert spaces (using measure theory): https://www.amazon.com/dp/0821890832/?tag=pfamazon01-20

Are you actually running into problems? You may find that it's not such a big deal.

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

Measure theory is a branch of mathematics that deals with the concept of measuring sets and functions. It is important for physics because it provides a rigorous framework for defining and analyzing physical quantities, which are often represented as sets or functions.

## What are the basic principles of measure theory?

The basic principles of measure theory include the concepts of measurable sets, measures, and integration. Measurable sets are those that can be assigned a numerical value, known as a measure, representing their size or extent. Integration is a mathematical operation that allows for the calculation of the total measure of a set.

## How does measure theory relate to probability theory?

Measure theory and probability theory are closely related, as they both deal with the concept of measuring sets and assigning numerical values to them. In fact, measure theory is often used as the foundation for probability theory, providing a rigorous framework for studying random events and their probabilities.

## Can you give an example of how measure theory is applied in physics?

One example of how measure theory is applied in physics is in the calculation of the volume of a three-dimensional object. By using the principles of measure theory, we can define the volume of the object as the integral of its measurable characteristics, such as its length, width, and height.

## Are there any limitations or challenges to using measure theory in physics?

While measure theory provides a powerful and rigorous framework for analyzing physical quantities, it can also be challenging to apply in certain situations. For example, in quantum mechanics, the concept of measurement is more complex and may require the use of more advanced mathematical techniques.

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