The general statement of the coarea inequality known also as the Eilenberg inequality is:

Theorem.If $f:X\to Y$ is a Lipschitz map between metric spaces and $A\subset X$, $0\leq m\leq n$, then $$ \int_Y^*\mathcal{H}^{n-m}(f^{-1}(y)\cap A)\, d\mathcal{H}^m(y)\leq (\operatorname{Lip} f)^m\frac{\omega_{n-m}\omega_m}{\omega_n}\mathcal{H}^n(A). $$

Federer proved it (Theorem 2.10.25 in [2]) under additional assumptions. Then Davies [1] showed that these additional assumptions are not necessary and a reader may find a detailed proof in [3].

For a related discussion, see Open problems in Federer's Geometric Measure Theory

My questions is that I am not exactly sure what are the assumptions in the theorem:

Question 1.Is it assumed here that the spaces $X$ and $Y$ are separable?

Question 2.Are $0\leq m\leq n$ any real numbers of just integers?

**[1] R. O. Davies,** Increasing sequences of sets and Hausdorff measure, *Proc. London Math. Soc.* 20(1970), 222-236.

**[2] H. Federer,** *Geometric measure theory.* Die Grundlehren der mathematischen Wissenschaften,Band 153 Springer-Verlag New York Inc., New York 1969.

**[3] L. P. Reichel,** The coarea formula for metric space valued maps, Ph.D. thesis, ETH Z ̈urich, 2009.
http://e-collection.library.ethz.ch/eserv/eth:289/eth-289-02.pdf.

**Edit.** $X$ and $Y$ can be any metric spaces and $0\leq m\leq n$ any real numbers. For a complete and detailed proof as well as historical account see:

**[4] B. Esmayli, P. Hajlasz**, The Coarea Inequality.
https://arxiv.org/abs/2006.00419.

The proof presented in [4] is elementary thanks to Fedja's help through Mathoverflow: Bounding an "integral" from below by the Hausdorff measure of the domain.