I dont understand what is the almost everwhere differentiable, can you give an example and definition please?
[Math] Almost everywhere differentiable definition
calculusreal-analysis
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They are of measure zero. It's because you're mistaking length with area. In $\mathbb R^2$, the Lebesgue measure is defined using the $\sigma$-algebra generated by rectangles over which the Lebesgue measure is just the area. You can therefore see that the infinite-length zero-area lines you speak of have measure zero.
Note that if you fix say $x_2$, the function $V(x_1) = |x_1| + |x_2|$ is locally Lipschitz and differentiable almost everywhere (i.e. except at $x_1 = 0$).
Hope that helps,
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.
Best Answer
Example:
Cantor's staircase.
It has derivative zero except on the Cantor set, which is a set of measure zero.