(approximately) the normalized area measure on the union of the large rectangles. The bad part, $\mu_{1, bad}$, is equal to $\mu_1 - {\mu_{1, good}}$, so it would be large on the slats and slightly negative on the parts of the large rectangles outside of the slats. If $T_0$ is the large rectangle containing $x$, then $d^x_* ({\mu_{1, good}}|_{T_0})$ would be much more spread out than $d^x_* (\mu_1 |_{T_0})$. On the other hand, if $T$ is far from $x$, then $d^x_* ({\mu_{1, good}}|_T)$ would be almost the same as $d^x_* (\mu_1 |_T)$. All together, the graph of $d^x_* ({\mu_{1, good}})$ would look like the graph of $d^x_* (\mu_1)$ with the peaks damped out. The pushforward $d^x_*({\mu_{1, good}})$ would be quite evenly spread and its $L^2$ norm would be finite. The graph of $d^x_*(\mu_{1, bad})$ would include the tall thin peaks from $d^x_*(\mu_1)$, and it would be slightly negative between the peaks. Since the thin peaks have small mass, the $L^1$ norm of $d^x_*(\mu_{1, bad})$ would be small.

To prove Theorem \[main\], we will show that the features of ${\mu_{1, good}}$ and $\mu_{1, bad}$ that we just observed in the example from Figure 1 will occur for any set $E$ of dimension 