As finite intersections of $(n-1)$-dimensional hypersurfaces, $E(P)$, $D(P)$ and $E(P)\cap D(P)$ you're right to say that will have $n$-dimensional Lebesgue measure zero. However, I don't think that helps you - I think the set you want to be small is the set of $P$s *with* bad points, not the set of bad-points themselves. Put a slightly different way, I think your probability space should be over $\Omega=(\mathbb{R}^{n\times n})^m$, not over $\mathbb{R}^n$ (c.f. Qfwfq's comment on your previous post).

I'm not sure that the $n$-dimensional Lebesgue measure of $E_P\cap \mbox{anything}$ is really such an interesting quantity if what you really care about is showing that tuples with $E_P\cap \{x: P_1x\wedge \dots P_mx=0\} \neq \emptyset$ are unstable - if $E_P$ is a (possibly empty) intersection of $(n-1)$-dimensional hypersurfaces, it's a given that it'll have $n$-dimensional measure $0$.

One other question: are your 'elipsoids' meant to be hollow (as the current definition suggests) or are they supposed to have interior? Put another way, are they 'hypersurfaces' or 'regions'?

Yes - I do mean $\{ x: \{P_1x,\dots,P_mx\} \;\mbox{linearly dependent}\}$. Typo followed by copy-and-paste ;)

By 'unstable' I mean that perturbing one of the 'bad' tuples (i.e. one of the ones for which the set $E_P \cap \{P_1x, \dots, P_mx\}$ of 'bad' points in nonempty) 'usually' gives you a 'good' tuple with no bad points.

I read your original problem like this: you have a way to associate a set $E_P$ with each $m$-tuple $P=(P_1,\dots,P_m)$ of $n\times n$ positive definite matrices, and you'd like to show that the tuples $P$ for which $E_P\cap \{P_1 x,\dots, P_m x\}\neq \emptyset$ are 'unusual' or 'unstable'. Is that a reasonable summary or am I missing the point?

Welcome to MathOverflow :) Can you provide a little more detail about the context of your problem? I think that in order to elicit helpful answers, you may need to specify e.g. what $\mathcal{H}$ is (I assume it's some Hilbert space of functions on $\mathbb{R}^d$), which loss functions you're considering, whether $\Omega$ is a semi-norm or something more exotic, whether you're interested in sufficient conditions, necessary conditions or both etc.

Upvoted because I like the paper :)

Also... what topology do you give to $C^1(E,E)$? Are you mainly interested in the finite dimensional case or do you need to allow $E$ infinite dimensional?

Re. $C^1(K,E)$ always being complete with the norm you suggest - I might be wrong, but I'm not sure that's even true for all compact subsets of $\mathbb{R}^d$, never mind when $E$ is something more exotic (it is true for all 'nice' compact subsets of $\mathbb{R}^d$ of course).

Re. $m$ and $n$: no that's a typo. Sorry! Re. what I'm getting at: how do you define the '$m$-factors' of an arbitrary element of the tensor product? (they aren't all of the form $x_1\otimes \dots \otimes x_m$). Note: this is possibly me just being ignorant, but it's possibly worth clarifying.

This is probably me being dumb, but... how does one define the $m$-factors of $\sum_{i\in I} x_{i_1}\otimes\dots \otimes x_{i_n}$ for an arbitrary finite $I$?

Ah - you fixed the $S_m$ :)

Could you clarify what you mean by $(\mathbb{C}^n)^{\otimes m}$ here? (I'm not sure I know what the "m-factors" of a generic element of $(\mathbb{C}^n)^{\otimes m}$ are, but perhaps I'm just naive). Also, I think the $S_n$ in your definition of $x_1\vee \dots \vee x_m$ should be an $S_m$.

Just for me (and others reading this), would you mind making your hypotheses concerning $X$, $Y$, $f$ and $g$ a little clearer? Is your hypothesis that $X$ and $Y$ are such that there is a $p>2$ such that $f(X), g(Y)\in L^p$ and (1) holds for all Lipschitz $f$ and $g$, or something else?

Re. I don't see where my logic fails, why do you think it does?