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Let $(X,d)$ be an $\mathcal{H}^n$-rectifiable metric space, i.e. there exits a collection of Lipschitz maps from measurable subsets of $\mathbb{R}^n$ to $X$ such that $ \mathcal{H}^n(X \backslash \cup …

Let $(X,d)$ be an arbitrary metric space and $E \subset X$ also arbitrary. Fix $s \in (0,\infty)$. Is it true that for any $ \delta > 0 $ and any collection of pairs $\{(A_i,a_i)\}_{i \in \math …

The theorem, as stated, is true for arbitrary metric spaces and for any pair of non-negative real numbers. Precisely, Theorem (Co-area Inequality). If $f:X\to Y$ is a Lipschitz map between any metric …

Let $\mathcal{M} \subset \mathbb{R}^N$ be an $n$-dimensional $C^1$ submanifold (connected). We have two metric functions on $\mathcal{M}$: The Euclidean distance inherited from $\mathbb{R}^N$. The i …

I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$. Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed …

I have the answer here: Fubini's Theorem on Arbitrary Foliations $$\int_U f = \int_{U_{\eta_0}} \left(\int_{U_\xi} f(\xi,\eta) \frac{|\det DG_{U_\xi} (\xi,\eta)| \cdot |\det DG_{U_{\eta_0}} (\xi,\eta …

The following is Corollary 7 of [1]. Theorem: For $X$ (an analytic subset of) a complete separable metric space, and $ s \in [0,\infty)$, the following is true about the Hausdorff measure $\mathcal{H …

So, the sought for paper is: Nöbeling, G., Hausdorffsche und mengentheoretische Dimension, Ergebnisse math. Kolloquium Wien 3, 24-25 (1931). And here is a ``translation" (to English and to modern ma …

In [1], Kirchheim proved the area formula for Lipschitz maps $f\colon \mathbb{R}^n\to X$ where $X$ is an arbitrary metric space, using the notion of metric differentiability. The metric derivative of …

In what follows $ \mathbb{R}^{n+m} = \{(x,y): x \in \mathbb{R}^n, \ y \in \mathbb{R}^m \} \ .$ Suppose $G: U \to V $ is a $C^1$-diffeomorphism from an open subset of a manifold to an open subset of …

There are several intersting consequences of this abstract viewpoint about these formulas. I also would have liked to discuss why the coarea inequality is the backbone of the coarea formula, but let m …