# Difference between Real Analysis and Probability Theory?

I do not really see a big difference between the two subjects. I was wondering if somebody can explain what the big difference between them is.

Let us compare the superficial differences:

1. In real analysis our subsets are called "measurable sets", in probability our subsets are called "events". The measure of a set in analysis is called the "measure", while in probability it is called "probability".
2. In real analysis we deal with "measurable functions", in probability theory we deal with "random variables".
3. In probability theory random variables induce "distributions", while in real analysis they are more naturally called "push-forwards".
4. In analysis we "integrate" with respect to the measure, in probability we compute the "expected value".
5. In analysis we say "almost everywhere" in almost every theorem, and in probability we say "almost surely" in almost every theorem.

There is one major difference:

• Probability theory assumes that we have a finite measure normalized to be equal to 1.

Other than that last part everything else seems to be essentially the same. It is the "finite measure assumption" which makes probability theory "work".

The only difference that I see is that, analysis is more general than probability theory. In mathematics we often require more generality with a compromise of some of its theorems. Is there something more?

• Strange question. The probability theory is built on measure theory and uses a lot of mathematical machinery from analysis. However, to call all real analysis a probability theory is ridiculous. Feb 11, 2015 at 4:10
• @Aksakal Why is it ridiculous when that is in fact true, unless I am missing something? Probability theory is a special case of analysis of finite measures. Feb 11, 2015 at 4:12
• If you want to be really strict, all of mathematics is set theory and logic... I think this question might fit better in a philosophy of science\mathematics community.
– Yair Daon
Feb 11, 2015 at 4:49
• @Richard No, the question is definitely not off topic. And no, it should not go to the mathematics board. The people there are mathematicians. The people here are statisticians and they will most likely have better responses. Feb 11, 2015 at 7:14
• @NicolasBourbaki, "Probability theory is a special case of analysis of finite measures" - that's one way to see it, and even if I agreed with you, your statement would contradict your question: a special case is not the same thing as the whole thing. However, I disagree with "special case" too. Probability theory is based on a subset of real analysis to fit into what we observe around us. It is not given that all math is "real". You configure parts of math to fit into reality and get fields such as probability theory. Feb 11, 2015 at 14:20

There is a huge difference. The key additions are the concepts of independence (of sigma-fields), conditional independence (given a sigma-field), and conditional expectation/probability (given a sigma-field), which don't play a central (if any) role in Real Analysis. Probability and Statistics without the concept of conditional independence are hardly possible, and definitely boring. In my opinion, Kolmogorovov's "Grundbegriffe der Wahrscheinlichkeitsrechnung" major contribution is the introduction of the general definition of conditional expectation (which depends on the Radon-Nikodym machinery). The importance of this concept in the development of modern Probability and Mathematical Statistics is hard to overstate. Take a look at "Probability with Martingales" by David Williams, and "Theory of Statistics" by Mark J. Schervish.

• +1 This answer gets to the heart of the matter. Analysis and probability have different interests: although they use similar sets of tools, they ask different questions and pursue almost completely different avenues of investigation. The two disciplines will nevertheless remain closely intertwined because insights from one can lead to progress in the other, much as (say) investigations of general relativity and quantum mechanics in physics have inspired advances in low-dimensional topology and operator theory, respectively. Feb 11, 2015 at 15:12
• I think mr Zen provided the appropriate answer I was looking for. Even though one can introduce independence and conditioning in analysis, it is not done. When I learned basic probability theory, I learned all the theorems of measures, integration, convergence, and decomposition. I never went so far into conditioning, which seems to be what makes probability theory unique. Feb 11, 2015 at 17:32
• Conditioning is the key. Since you already have training in analysis, read Williams. You will love it. Fascinating book.
– Zen
Feb 11, 2015 at 17:49
• Nice little glitch in the SE technology: I have been able to upvote this answer twice--once on CV and once again here :-). Feb 11, 2015 at 18:13
• There is also the interpretation of $\sigma$-algebras as "accessible information" in probability, which is quite natural in light of the conditioning framework and very useful in stochastic processes. Analysis doesn't tend to use this idea. That said, I will remark that conditional expectation does not necessarily depend on the usual machinery used to prove Radon-Nikodym. One can use for example just Hilbert space methods to formulate it.
– Ian
Feb 11, 2015 at 19:05

Maybe we can put the question in the context of a "mathematical model". Kolmogorov (and others) came up with a model for probability theory that involves a measure in the sense of Lebesgue. Great. Now we can use the tools of measure theory to study probability theory. But certainly there is no reason to call them "the same".

Similarly, in physics, there have been given certain mathematical models for phenomena in the real world. But it is important not to confuse the model with the phenomenon modeled.

Here is a quote I like:

THESIS 22: Those who seek a phenomenon which exactly follows a mathematical model, seek in vain.

(F. Topsoe, Spontaneous Phenomena, Academic Press, 1990)

Zen answered the question: "what does probability theory have that real analysis doesn't"? There is a corresponding question: What does real analysis have that probability theory doesn't?

And one possible answer to that is:

• metric spaces
• general notions of spaces of functions (of which stochastic processes are an example, but not the entirety)
• calculus, including the derivative
• analysis of functions in terms of orthonormal bases -- probability theory uses some of these tools (mgf, probability generating function, etc), but a course in probability typically does not develop the machinery in its most general form.
• different notions of integration than Lebesgue
• different applications, especially in classical physics

This is just a former undergrad's experience. More experienced students will have more examples.

Just came across this question and know it's years past being asked, but couldn't resist sharing this old blogpost of Gowers' that I read a few days back that illuminated my understanding on this very question. He talks about the Berkeley probabilist David Aldous speaking at ICM 2010. It's quite long so I won't bother excerpting it. Here's the link though, well worth your time.

https://gowers.wordpress.com/2010/09/01/icm2010-fourth-day/

The gist -- as I interpret it-- is, from the probabilistic perspective, we don't care so much for the function's domain/sample space and care more about how it's realized, that is, its distributional properties (and this makes sense because we can show that any random variable with distribution specified can be defined with $$[0,1]$$ as the sample space; think binary expansions modelling random walk sample paths). And that in turn becomes important when we consider independent copies of the same random variable and the possibility of their co-existence (that is, we can find non-constant $$f$$ and $$g$$ continuous and independent on ($$[0,1]$$, Borel, Leb). Hint, think space filling curves on $$[0,1]^2$$ with product measure.). This is particularly important from the statistical application perspective, almost philosophically so one might say, because it allows us to consider the questions of sampling and simulation.

Measure theory or real and complex analysis is more general than probability theory because probability theory when based on measure theory is just an application area.

But there are other possibilities for foundations of probability theory, because C* - algebra could also be used as a foudation for constructing familiar structures for mathematical statisticians

http://jointmathematicsmeetings.org/meetings/national/jmm2013/AMSColloquium.pdf

• Well, yes non-commutative probability is another model, and it is particularly useful in (quantum) physical models. However I haven't seen it used much for general statistics. I agree with your overall point though :)
– P.Windridge
Feb 11, 2015 at 11:12
• @Analyst You should not compare real and complex analysis. Complex analysis is not just real analysis with a complex component, the subjects are radically different. For one thing, there is hardly any mention of measure theory within complex analysis. Feb 11, 2015 at 17:34