Let $X$ be the Thomas plank as defined in the body of the question. Let $\mathcal{Z}(X)$ be the set of zero sets of $X$ i.e. sets of zeros of continuous maps $f\in C(X)$. Lemma. If $f\in C(X)$ and $Z(f) = \{x\in X : f(x) = 0\}$, then $Z(f)\cap L_0$ is countable or co-countable in $L_0$. Proof: Ac...

Let $f:X\times X\rightarrow R$ be integrable let $g_1,g_2:X\rightarrow X\times X$ be two measurable functions and $\mu$ a measure on $X$. Define a measure $\eta$ on $X\times X$ by $\eta(A\times B)= \mu(g_1^{-1}(A)\cap g_2^{-1}(B))$ Show that $\int f(x,x') d\eta(x,x')=\int f(g_1(x),g_2(x))d\mu(x)...