Meaning of measure zero, and exclusion of set in the Filippov set valued map (Regularization) (Differential Inclusion)F[X]

Question

The Filippov set-valued map (regularization) form in https://arxiv.org/pdf/0901.3583.pdf equation 19

is

$$ F[X(x)] = \bigcap_{\delta>0}\bigcap_{\mu(S)=0} \bar{co}\{X(B(x,\delta)\setminus S)\}\\ B(x,\delta) = \{v \in \mathbb{R} | ||x-v||<\delta\}\\ \mu(S) = \text{Lebesgue Measure zero}\\ \bar{co} = \text{convex closure (Hull)}\\ $$

I understood how Filippov set-valued map of the above equation is calculated through simple examples but I quite don't understand what the above equation is translated.

say

$$ X(x) = -sign(x) = \begin{cases} -1, & x > 0 \\ 0, & x = 0 \\ 1, & x < 0 \end{cases} $$

applying the Filippov set-valued map, becomes $$ F[X(x)] = \begin{cases} -1, & x > 0 \\ [-1,1], & x = 0 \\ 1, & x < 0 \end{cases} $$

I do not understand what the $\mu(S)=0 $ and why used $\{B(x,\delta)\setminus S\} $ and the inclusion notation $\bigcap$.

1. The reference implies that the $\mu(S)=0 $ is a point or set with discontinuity. In this case, it would be the point with $x=0$. Is it correct to think the Lebesgue measure zero sets are the set with discontinuity?

2. I assume that the set S is $ \{ \{x>0\}, \{x=0\}, \{x<0\} \} $ then $\mu(S)=0 $ is a set when $\{x=0\}$ then why when forming the convex hull, the set S is excluded from the open ball of ($\{B(x,\delta)\setminus S\} $) Shouldn't it also include the set S? Even in the answer above (F[X(x)]), [-1,1] is defined on x=0 which should have been excluded (I think) if $\{B(x,\delta)\setminus S\} $ means excluding the set S from $B(x,\delta)$.

3. If set S is excluded when forming the the convex closure (hull), why F[X(x)] includes functions of the set $ x > 0 $ and $ x < 0 $? Should this only includes only $ F[X(x)] = [-1,1], x = 0 $?

4. And why use the intersection $\bigcap$ of the set, not union?

I only have undergraduate math knowledge. Please help me how to interpret this Filippov notation. Thanks in advance.

Answer 1

The meaning of the set of measure 0 is explained in the text your provided (page 13). For simplicity this is the paragraph of interest:

The mathematical framework for formalizing this neighborhood idea uses set-valued maps. The idea is to associate a set-valued map to $X:\mathbb{R}^d\mapsto\mathbb{R}^d$ by looking at the neighboring values of $X$ around each point. Specifically, for $x\in\mathbb{R}^d$, the vector field $X$ is evaluated at the points belonging to $B(x,\delta)$, which is the open ball centered at $x$ with radius $δ>0$. We examine the effect of $δ$ approaching 0 by performing this evaluation for smaller and smaller $δ$. For additional flexibility, we exclude an arbitrary set of measure zero in $B(x, δ)$ when evaluating $X$, so that the outcome is the same for two vector fields that differ on a set of measure zero.

The use of the intersection is considered to take the smallest set of interest at the discontinuities. You can think of the intersection of sets to be analogous to taking the minimum.

For instance, if you consider:

$X(x) = \left\{\begin{array}{llcl} -1-x, & x>0\\ 0,& x=0\\ 1-x,& x<0 \end{array}\right.$

The intersection makes sure that you only consider the values very near the discontinuity. Taking the union for all $\delta>0$ would take too many values around the discontinuity.