Probability with measure theory

In summary, the conversation is about finding a book for studying probability theory using measure theory. The person is taking their first course in probability and is looking for a recommendation on a book that develops the theory of probability using measure theory, suitable for self-study. One recommendation given is "A First Look at Rigorous Probability Theory" by J.S. Rosenthal, which may require some prior knowledge in probability. Another suggestion is Resnick's "A Probability Path". The person will consider both options.
  • #1
mr.tea
102
12
Hi,

I am looking for a book for studying probability theory using measure theory. This is the first course I am taking of probability. Notions and theorems from measure theory are part of this course.
As it turns out, this is a catastrophic disaster, and the textbook for this course is also not helping a lot(and doesn't even use measure theory).
Therefore I need a recommendation on a book that develops the theory of probability using measure theory, and if it is possible, suitable for self study.

Thank you.
 
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  • #2
Nice question. I think the book "A First Look at Rigorous Probability Theory" by J.S. Rosenthal may be suitable for this purpose. It does not contain enough material to serve as a long-time reference, but it does a very good job introducing the subject.
 
  • #3
Krylov said:
Nice question. I think the book "A First Look at Rigorous Probability Theory" by J.S. Rosenthal may be suitable for this purpose. It does not contain enough material to serve as a long-time reference, but it does a very good job introducing the subject.
Thank you for the answer. But unfortunately it seems that the book assumes some knowledge in probability.
 
  • #4
mr.tea said:
Thank you for the answer. But unfortunately it seems that the book assumes some knowledge in probability.
Most of the times people first take a non-measure-theory based course on probability. Then for a second course everything is placed in the proper measure-theoretic context. So, I conjecture that it will be hard to find a "first course in probability" based on measure theory.

With that being said, I believe that Rosenthal's book can be read by someone who has no prior exposure to probability. It may be more important that you have an understanding of introductory analysis and some experience with proof writing. Another title you could consider is Resnick's "A Probability Path".
 
  • #5
Krylov said:
Most of the times people first take a non-measure-theory based course on probability. Then for a second course everything is placed in the proper measure-theoretic context. So, I conjecture that it will be hard to find a "first course in probability" based on measure theory.
This course is called "Basic..", and suppose to be the first course in probability that we should take(math major).

With that being said, I believe that Rosenthal's book can be read by someone who has no prior exposure to probability. It may be more important that you have an understanding of introductory analysis and some experience with proof writing. Another title you could consider is Resnick's "A Probability Path".

I will give Rosenthal's book another chance, and also look at the other book. Thank you.
 

1. What is the difference between probability and measure theory?

Probability theory is a branch of mathematics that studies the likelihood of events occurring. It deals with random variables, discrete and continuous distributions, and various principles such as the law of large numbers and the central limit theorem. Measure theory is a more general mathematical framework that provides a foundation for probability theory. It deals with the concept of measure, which assigns a numerical value to sets in a given space. Probability theory can be seen as a special case of measure theory, where the measure is defined on the set of all possible outcomes of a random experiment.

2. How is measure theory used in probability theory?

Measure theory is used in probability theory to define the probability of events in a rigorous and consistent way. The measure of a set represents the likelihood of that set occurring in a given space. By defining a measure on a set of possible outcomes, we can determine the probability of different events occurring and make predictions about the behavior of random variables.

3. What is a probability measure?

A probability measure is a function that assigns a numerical value, between 0 and 1, to a set of possible outcomes in a given space. This function must satisfy certain axioms, such as the measure of the entire space being equal to 1 and the measure of the empty set being equal to 0. Probability measures are used to determine the likelihood of events in probability theory.

4. What is the difference between discrete and continuous probability measures?

A discrete probability measure is one that assigns probabilities to individual outcomes that can be counted, such as the probability of rolling a specific number on a die. A continuous probability measure, on the other hand, assigns probabilities to intervals of values, such as the probability of a randomly chosen person being between a certain height range. In measure theory, both discrete and continuous measures can be defined and used to model different types of events.

5. How is measure theory used in real-world applications?

Measure theory has many real-world applications, particularly in fields such as statistics, economics, and physics. In statistics, measure theory is used to define and analyze probability distributions, which are essential for making predictions and drawing conclusions from data. In economics, measure theory is used to model and analyze uncertain events, such as stock market fluctuations. In physics, measure theory is used to define and study the behavior of quantum systems, which have inherent uncertainty and randomness.

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