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Can anyone help me with my big problem about real analysis and measure theory. In Mathematics are a lot of theorem which are proved with the help of the notion $\mu$-almost everywhere . I know that $\mu$ is a measure, but I want to know when can I apply the theory of $\mu$-almost everywhere and when I cannot. Can you give me some example please.

Thanks :)

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A property of elements of a measure space holds $\mu$-almost everywhere if there is a measurable set $N$ with measure zero such that the property holds for all elements of the measure space that are not in $N$. –  Michael Greinecker Jan 28 '13 at 13:27
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Well, to make things more messy I would say that $\mu$-a.e. results are $\mu$-relatively strong. Namely, such results do not hold on sets of very small important (of $\mu$ measure zero). It all depends on the measure $\mu$ in the end, and on how do you rely upon it. Namely, measures are weights that tell you what is more important and what is less.

If you know that

a bounded function $f:[0,1]\to\Bbb R$ is Riemann integrable iff the set of discontinuities $D(f)$ is of the Lebesgue measure zero

then it gives you a pretty nice characterization of those functions that are Riemann integrable. Indeed, you have both a cool intuition that such functions does not have to be very weird (their set of weirdness $D(f)$ is "small enough") and you have a working criteria to check integrability in all particular cases.

If you know that

the Poisson process has a finite number of jumps $\mathsf P$-a.e. (where $\mathsf P$ is a probability measure)

you will think, that simulating a trajectory of such process with exact random number generator will be successful in your whole entire life.

However, in case you are able to show that something holds $\delta_x$-a.e. where $\delta_x$ is a Dirac measure (a point mass) at a single point $x$, this is not a very useful result usually. Indeed, it tells you that some property holds at a point $x$ and does not give you any further information. Agree?

Well, as I mentioned above it's all relative and in fact some single point $x$ may be more important for you then the rest of the state space. For example,

the set $A$ is visited by a $\psi$-irreducible Markov Chain at least once regardless of the initial condition iff $\psi(A)>0$.

Here the measure $\psi$ tells us which sets are "big enough" w.r.t. the dynamics of a Markov Chain. Namely, which sets, no matter where do you start from, you'll visit with a positive probability. Such sets however do not have to be "big" in our usual understanding. Namely, it may happen naturally happen for some probabilistic models that $\psi(\{0\}) = 1$ and $\psi(\Bbb R\setminus \{0\}) = 0$. Does it mean that the singleton set $\{0\}$ is bigger than $\Bbb R\setminus \{0\}$? Relatively, yes - under these particular conditions the behavior of a process at $\{0\}$ determines the whole asymptotic behavior of it.

Hope, these examples help.

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To be strictly correct, you'll want to change "the function $f:[0,1]\to\Bbb R$ is Riemann integrable iff" to "the bounded function $f:[0,1]\to\Bbb R$ is Riemann integrable iff". –  Dave L. Renfro Jan 28 '13 at 17:07
@DaveL.Renfro: thanks for the typo –  Ilya Jan 28 '13 at 18:37
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