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Singular Radon probabilities on $[0,1]^d$. Is conditioning on $x_i = \alpha$ well-defined?

The question: Let $\pi$ be a Radon probability measure on $[0,1]^d$, $2\leq d < \omega$, that is singular (w.r.t. to the $d$-dimensional Lebesgue measure). Suppose that for $i\in \{1,\dots,d\}$ and $\ …