Dependencies

If the \(A_i\) and \(B_i\) are topological spaces and the \(\phi _i\) are continuous functions, then the restricted product \(\phi = \prod '_i\phi _i\) is a continuous function.

Say \(R\) and \(S\) are topological rings, and \(S\) is an \(R\)-algebra, finite as an \(R\)-module. Assume that the topology on \(S\) is the \(R\)-module topology. Now say \(M\) is an \(S\)-module, and give it the induced \(R\)-module structure. Then the \(R\)-module topology and \(S\)-module topology on \(M\) coincide.

Say \(B\) is a finite-dimensional central simple algebra over a field \(k\), and \(u\in B^\times \). Let \(\ell _u:B\to B\) be the \(k\)-linear maping \(x\) to \(ux\) and let \(r_u:B\to B\) be the \(k\)-linear map sending \(x\) to \(xu\). Then \(\det (\ell _u)=\det (r_u)\).

If \(B\) is a central simple algebra over a locally compact field \(F\), and if \(u\in B^\times \), then \(d_B(r_u)=\delta _B(u)\) (recall that the latter is defined to mean \(d_B(\ell _u)\)).

If \(A\) is a locally compact topological additive abelian group, if \(\mu \) is a regular Haar measure on \(A\), and if \(\phi :A\to A\) is an additive homeomorphism, then we let \(d_A(\phi )\) denote the unique positive real number such that \(\mu (X)=d_A(\phi )(\phi _*\mu )(X)\) for any Borel set \(X\).

If \(\mu \) is any regular Haar measure on \(A\) then \(d_A(\phi )\mu = \phi ^*\mu .\)

\(d_A(\phi )\) is independent of choice of regular Haar measure.

If \(f:A\to B\) is a group homomorphism and open embedding between locally compact topological additive groups and if \(\alpha :A\to A\) and \(\beta :B\to B\) are additive homeomorphisms such that the square commutes (i.e., \(f\circ \alpha =\beta \circ f\)) then \(d_A(\alpha )=d_B(\beta )\).

\(d_V(\phi )=\delta _F(\det (\phi ))\), where \(\det (\phi )\in F\) is the determinant of \(\phi \) as an \(F\)-linear map.

If \(\mu \) is any regular Haar measure on \(A\) then \(d_A(\phi )(\phi _*\mu ) = \mu .\)

If \(A_i\) are a finite collection of locally compact topological abelian groups, with \(\phi _i:A_i\to A_i\) additive homeomorphisms, then \(d_{\prod _i A_i}(\prod _i\phi _i)=\prod _i d_{A_i}(\phi _i)\).

If \((A,+)\) and \((B,+)\) are locally compact topological abelian groups, and if \(\phi :A\to A\) and \(\psi :B\to B\) are additive homeomorphisms, then \(\phi \times \psi :A\times B\to A\times B\) is an additive homeomorphism (this is obvious), and \(d_{A\times B}(\phi \times \psi )=d_A(\phi )d_B(\psi )\).

\(d_A(id)=1.\)

With \(A\), \(A_i\), \(C_i\), \(\phi _i\), \(\phi \) defined as above, we have \(\delta _A(\phi )=\prod _i\delta _{A_i}(\phi _i)\).

If \(f:A\to \mathbb {R}\) is a Borel measurable function then \(d_A(\phi )\int f(x)d\mu (x)=\int f(x)d\phi ^*\mu (x)\).

If \(f:A\to \mathbb {R}\) is a Borel measurable function then \(d_A(\phi )\int f(x)d\phi _*\mu (x)=\int f(x)d\mu (x)\).

If \(X\) is a Borel set then \(\mu (X)=d_A(\phi )\mu (\phi ^{-1}X)\).

\(d_A(\phi \circ \psi )=d_A(\phi )d_A(\psi ).\)

In the above situation (\(V\) a finite-dimensional \(\mathbb {Q}\)-vector space, \(\phi :V\cong V\) is \(\mathbb {Q}\)-linear, \(\phi _{\mathbb {A}}\) the base extension to \(V_{\mathbb {A}}:=V\otimes _{\mathbb {Q}}{\mathbb {A}_{\mathbb {Q}}}\), a continuous linear endomorphism of \(V_{\mathbb {A}}\) with the \(\mathbb {A}_{\mathbb {Q}}\)-module topology), we have \(d_{V_{\mathbb {A}}}(\phi _{\mathbb {A}})=1.\)

If \(u\in R^\times \) then \(\delta _R(u)=\delta _F(\det (\ell _u))\).

Say \(A\) is a compact topological additive group and \(\phi :A\to A\) is an additive isomorphism. Then \(d_A(\phi )=1.\)

We define \(\delta _R(u)\) (or just \(\delta (u)\) when the ring \(R\) is clear) to be \(d_R(\ell _u)\).

If \(x\in \mathbb {A}_{\mathbb {Q}}^\times \) then \(\delta _{\mathbb {A}_{\mathbb {Q}}}(x)=\prod _v|x_v|_v.\)

If \(x\in \mathbb {Q}^\times \subseteq \mathbb {A}_{\mathbb {Q}}^\times \) then \(\delta _{\mathbb {A}_{\mathbb {Q}}}(x)=1.\)

If \(R=\mathbb {C}\) then \(\delta _R(u)=|u|^2\).

The function \(\delta _R:R^\times \to \mathbb {R}_{{\gt}0}\) is continuous.

If \(f:R\to \mathbb {R}\) is a Borel measurable function and \(u\in R^\times \) then \(\delta _R(u)\int f(ux)d\mu (x)=\int f(x)d\mu (x)\).

If \(X\) is a Borel subset of \(R\) and \(r\in R^\times \) then \(\mu (rX)=\delta _R(r)\mu (X)\).

If \(R=\mathbb {Q}_p\) then \(\delta _R(u)=|u|_p\), the usual \(p\)-adic norm.

If \(R_i\) are a finite collection of locally compact topological rings, and \(u_i\in R_i^\times \) then \(\delta _{\prod _i R_i}((u_i)_i)=\prod _i\delta _{R_i}(u_i)\).

If \(R\) and \(S\) are locally compact topological rings, then \(\delta _{R\times S}(r,s)=\delta _R(r)\times \delta _S(s)\).

If \(R=\mathbb {R}\) then \(\delta _R(u)=|u|\).

If \(u=(u_i)_i\in R^\times \) then \(\delta _R(u)=\prod _i\delta _{R_i}(u_i)\).

If \(B\) is a finite-dimensional \(\mathbb {Q}\)-algebra and if \(b\in B^\times \) then right multiplication by \(b\) does not change Haar measure on \(B_{\mathbb {A}}\).

Let \(B\) be a finite-dimensional central simple \(K\)-algebra. Say \(u\in B_{\mathbb {A}}^\times \), and define \(\ell _u\) and \(r_u:B_{\mathbb {A}}\to B_{\mathbb {A}}\) by \(\ell _u(x)=ux\) and \(r_u(x)=xu\). Then \(d_{B_{\mathbb {A}}}(\ell _u)=d_{B_{\mathbb {A}}}(r_u)\).

If \(K\) is a number field and \(V\) is an \(K\)-module, then the natural isomorphism \(V\otimes _K\mathbb {A}_K=V\otimes _{\mathbb {Q}}\mathbb {A}_{\mathbb {Q}}\) induced by the natural isomorphism \(\mathbb {A}_K=K\otimes _K\mathbb {A}_{\mathbb {Q}}\) is a homeomorphism if the left hand side has the \(\mathbb {A}_K\)-module topology and the right hand side has the \(\mathbb {A}_{\mathbb {Q}}\)-module topology.

If \(B\) is a finite-dimensional \(\mathbb {Q}\)-algebra (for example a number field, or a quaternion algebra over a number field), if \(B_{\mathbb {A}}\) denotes the ring \(B\otimes _{\mathbb {Q}}\mathbb {A}_{\mathbb {Q}}\), and if \(b\in B^\times \), then \(\delta _{B_{\mathbb {A}}}(b)=1\).

Let \(A\) and \(B\) be locally compact topological groups and let \(f:A\to B\) be both a group homomorphism and open embedding. The pullback along \(f\) of a Haar measure on \(B\) is a Haar measure on \(A\).

The pullback of a regular Borel measure along an open embedding is a regular Borel measure.