My question is very direct: considering the parametric space $\Theta = \mathbb{R}^n$, n>3, I want to know if the set $ \{ (x_1, \ldots, x_n) \in \mathbb{R^n}: x_1 = x_2 = x_3\} $ has Lebesgue measure zero in $\Theta$?
My intuition say "Yes", for the same reason pointed here: $\mathbb{R}^n\times\{0\}$ has measure zero in $\mathbb{R}^{n+1}$
But, I don't know for sure. If my intuition is correct, I would really like that someone show me the path to rigorously demonstrate it.
Apply Sard's theorem to the linear map $(x_1, x_2, x_3, x_4, \ldots, x_n) \mapsto (x_1, x_1, x_1, x_4, \ldots x_n)$, which has rank zero, and the claim follows immediately.
Of course, this is a bit like killing a fly with a sledgehammer --- chances are good that if you're stuck on this particular claim, you've not yet encountered Sard's theorem. But in the rare cases when I want to prove something's neglible, Sard is the go-to tool in my toolbox.