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: 


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*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". 

*In real analysis we deal with "measurable functions", in probability theory we deal with "random variables". 

*In probability theory random variables induce "distributions", while in real analysis they are more naturally called "push-forwards". 

*In analysis we "integrate" with respect to the measure, in probability we compute the "expected value". 

*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? 
 A: 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.
A: 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)
A: 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.
A: 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.  
A: 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
