The "intersection property" of the symmetric difference metric

Question

$\newcommand{\measure}{\operatorname{measure}}$

The symmetric difference between sets can be used to define a pseudo-metric on the set of subsets of a given measure space:

$$d(S,T)=\measure(S\oplus T)$$

This pseudometric has the following property, that can be called "Intersection Property":

If $S$ intersects $T$, then there is a constant $r>0$ such that every set $S'$ with $d(S',S)<r$ intersects every set $T'$ with $d(T',T)<r$ [where "$S$ intersects $T$" means: $d(S\cap T, \phi)>0$].

Proof: Define $r = d(S\cap T, \phi)/3 = \measure(S\cap T)/3$. By assumption $r>0$. The set $S\cap T$ can be decomposed as follows:

$$S\cap T = $$ $$[(S\setminus S')\cap(T\setminus T')] \cup $$ $$[(S\cap S')\cap(T\setminus T')] \cup $$ $$[(S\setminus S')\cap(T\cap T')] \cup $$ $$[(S\cap S')\cap(T\cap T')]$$

By definition, $S\oplus S' = (S\setminus S')\cup (S'\setminus S)$. Hence:

$$\measure(S\setminus S')\leq \measure(S\oplus S') < r$$

Simlarly:

$$\measure(T\setminus T')\leq \measure(T\oplus T') < r$$

Substituting in the decomposition of $S\cap T = $ gives:

$$\measure[(S\cap S')\cap(T\cap T')] > \measure[S\cap T] - 3r > 0$$

This implies that $S$ intersects $S'$ and $T$ intersects $T'$.

My questions are:

A. Is this proof correct?

B. Is there a formal term for this property of pseudo-metrics, which I called "intersection property"?

NOTE: This property is not true for all pseudo-metrics. For example, it is not true for the pseudo-metric based on the Hausdorff distance.

Answer 1

A. Almost correct. At the end you say that $\text{measure}[S\cap T]-3r>0$, but actually it equals zero. Also at the end, you conclude that $S$ intersects $S'$ and $T$ intersects $T'$, but that's not what you're trying to prove: what you want is that $S'$ intersects $T'$. Happily, this also follows from the previous line.

B. I don't know of a standard term for this property. It does, however, follow from the intersection operation being (jointly) continuous at $S\cap T$, which is a pretty standard property. The property I mean here is that for every $\epsilon>0$ there exists $\delta>0$ such that for any sets $S'$ and $T'$ satisfying $d(S,S')<\delta$ and $d(T,T')<\delta$, we have $d(S\cap T,S'\cap T')<\epsilon$. This holds for the symmetric difference metric; in fact, $$ d_\oplus(S\cap T,S'\cap T') \le d_\oplus(S,S')+d_\oplus(T,T') $$

As for the Hausdorff metric, here are a few related results:

1. Intersection is "upper semicontinuous" for nonempty closed and bounded sets in a metric space. Precisely: if $A,A',B,B_n$ are nonempty closed and bounded sets in a metric space $X$, and $B_n\to B$ in Hausdorff metric, and $B_n\cap A=A'$ for all $n$, then $B\cap A\supseteq A'$.

2. The inclusion in part 1 may be strict: for example, take $X=\mathbb{R}$, $A=\{0,1\}$, $A'=\{1\}$, and $B_n=\{\frac1n,1\}$.

3. Intersection is jointly continuous for closed convex sets in $\mathbb{R}^d$, at points where the intersection has interior. Precisely: if $K,K_n,L,L_n$ are closed convex sets in $\mathbb{R}^d$, and $K_n\to K$ and $L_n\to L$ in Hausdorff metric, and $\text{int}(K\cap L)\ne\varnothing$, then $K_n\cap L_n\to K\cap L$.