Let $B \subset \mathbb R^n$ be the unit ball centered at the origin, $I = [0,1] \subset \mathbb R$, and $C=B\times I\subset \mathbb R^{n+1}$ be a cylinder.
Let $\chi_C$ be a the indicator function of the cylinder. By Tonelli's theorem: $$ |C| = \int_C 1\, dx = \int_I \left ( \int_B 1 \,d(x_1,...,x_n) \right) dx_{n+1} = \mathcal H^n(B). $$
What I am wondering is what happens if we change $(n+1)$-Lebesgue measure for $n$-dimensional Hausdorff measure.
Let $E$ be a set inside $C$ with finite $n$-Hausdorff measure (imagine an $n$-dimensional smooth compact manifold or the graph of a $C^1$ function). Can I say (I definitely need further assumptions on the set $E$, but I am not sure which ones) that $$ \mathcal H^n(E) = \int_0^1 \mathcal H^{n-1}(E \cap B\times\{t\}) dt $$ or $$ \mathcal H^n(E) = \int_B \operatorname{card}\left(\{E\cap (\{x\}\times I)\}\right) d\mathcal H^n(x)? $$ I can see that one possible obstruction might be that $E$ could be an horizontal (or vertical) $n$-dimensional set (say $E = B \times \{0\}\subset \mathbb R^{n+1}$). It looks like these decompositions are almost never true, but is there any way to fix it? Does anyone have some reference on this kind of decomposition of measures or related things and techniques?
I'm sorry if this question is not very well formulated, I am not sure about how to state my doubt. If somebody wants to help, I'd be happy to try to give further clarification to what I mean.
I don't think your first formula holds. Take $E = B \times \{1/2\}$ for example. ${\cal H}^n(E)$ is positive. On the other hand, the function under the integral is zero for all $t \neq 1/2$ so its integral is zero.
I guess that what you are interested in is geometric measure theory and integral geometry. If instead of slicing in one direction, you average the measure on the slices in all possible directions, then the formula holds. The simplest example of such result is Crofton formula.