 Solution

Take independent random variables $U_1,\dots ,U_d$, each uniform on $[0,1/2]$, and independent Rademacher random variables ${\epsilon }_1,\dots ,{\epsilon }_{d-1}$. Defineand let $\eta$ be the joint law of $(X_1,\dots ,X_d)$.

Each $X_i$ is uniform on $[-1/2,1/2]$. If the $d$th coordinate is omitted, the remaining signs are the independent ${\epsilon }_i$. If coordinate $j<d$ is omitted, the remaining signs arewhich are again uniformly distributed over all $2^{d-1}$ sign vectors because this map from $({\epsilon }_1,\dots ,{\epsilon }_{d-1})$ is a bijection. The independent magnitudes then show that every $(d-1)$-coordinate marginal is Lebesgue measure.

The full law is not $d$-dimensional Lebesgue measure, becausealmost surely, whereas this product is equally likely to have either sign under independent Lebesgue coordinates.