91Ó°ÊÓ

If \(\mu\) is a Radon measure on the locally compact group \(G\) and \(f \in C_{c}(G)\), the functions \(x \rightarrow \int L_{x} f d \mu\) and \(x \rightarrow \int R_{x} f d \mu\) are continuous.

Let \(\mathbb{Q}\) have the relative topology induced from \(\mathbb{R}\). Then \(\mathbb{Q}\) is a topological group that is not locally compact, and there is no nonzero translation-invariant Borel measure on \(\mathbb{Q}\) that is finite on compact sets.

If \(f:(a, b) \rightarrow \mathbb{R}^{n}\) is a parametrization of a smooth curve (i.e., a 1-dimensional \(C^{1}\) submanifold of \(\mathbb{R}^{n}\) ), the Hausdorff 1-dimensional measure of the curve is \(\int_{a}^{b}\left|f^{\prime}(t)\right| d t .\)

If \(\phi: \mathbb{R}^{k} \rightarrow \mathbb{R}\) is a \(C^{1}\) function, the graph of \(\phi\) is a \(k\)-dimensional \(C^{1}\) submanifold of \(\mathbb{R}^{k+1}\) parametrized by \(f(x)=(x, \phi(x))\). If \(A \subset \mathbb{R}^{k}\), the \(k\)-dimensional volume of the portion of the graph lying above \(A\) is $$ \int_{A} \sqrt{1+|\nabla \phi(x)|^{2}} d x $$ (First do the linear case, \(\phi(x)=a \cdot x\), Show that if \(T: \mathbb{R}^{k} \rightarrow \mathbb{R}^{k+1}\) is given by \(T x=(x, a \cdot x)\), then \(T^{*} T=I+S\) where \(S x=(a \cdot x) a\), and hence \(\operatorname{det}\left(T^{*} T\right)=\) \(1+|a|^{2}\). Hint: The determinant of a matrix is the product of its eigenvalues; what are the eigenvalues of \(S ?\) )

In any metric space, zero-dimensional Hausdorff measure is counting measure.