Neukirch, Haar measure on Minkowski space
$begingroup$
I would like to understand what Neukirch means when he writes down
$$ vol_{canonical}(X) = 2^s vol_{Lebesgue}(f(X)) $$
(Neukirch, Algebraic Number Theory. Pg 31)
I will write down the details at the end of the post.
I think it suffices to understand the following situation:
Let $V$ be a Euclidean vector space.
This means that $V$ is a finite-dimensional real vector space with symmetric, positive-definite bilinear form $langle ~,~ rangle : V times V to mathbb{R}$. Let $mu$ be the Haar measure of a Euclidean vector space (such that the cube spanned by the orthonormal basis has measure 1).
Let $e_i in mathbb{R}^n$ be the vector with $1$ on $i$th coordinate and $0$ everywhere else. Let $Phi$ be the cube spanned by $e_i$. Then $mu_{Leb}(Phi)=1$ where $mu_{Leb}$ is the Lebesgue measure on $mathbb{R}^n$.
Question: Let $f: V to mathbb{R}^n$ be a linear isomorphism such that $$ mu(f^{-1}(Phi)) = c $$ or equivalently $$ mu(f^{-1}(Phi))= c mu_{Leb}(Phi) $$ Then for any measurable set $X subset V$, $$ mu(X) = c mu_{Leb}(f(X)) $$
Details: Let $K/mathbb{Q}$ be a number field with $[K:mathbb{Q}]=n$. Let $K_mathbb{R}$ be the set of $n$-tuple of $mathbb{C}$ indexed by $Hom(mathbb{Q}, mathbb{C})$ with the following properties. If $rho_1, ldots, rho_r$ are real embeddings of $K$ and $tau_1, overline{tau_1}, ldots, tau_s, overline{tau_s}$ are the complex embeddings, then elements $(z_sigma)$ in $K_mathbb{R}$ is of the form
$$ z_rho in mathbb{R} text{ and } z_overline{tau} = overline{z_tau} $$
where $sigma$ varies over $Hom(K,mathbb{C})$. $K_mathbb{R}$ becomes a finite-dimensional real vector space, so it has a Haar measure $vol_{canonical}$. Let $f: K_mathbb{R} to mathbb{R}^n$ be a linear isomorphism defined by $(z_sigma) mapsto (x_sigma)$ where
$$ x_rho = z_rho text{ and } x_tau = Re(z_tau), x_overline{tau} = Im(z_tau) $$
$K_mathbb{R}$ has a symmetric, positive-definite bilinear form
$$ langle z, w rangle = sum_sigma z_sigma overline{w_sigma} $$
which transfers to $mathbb{R}^n$ by
$$ (x,y) = sum_sigma alpha_sigma x_sigma y_sigma $$
where $alpha_rho = 1$ for real $rho$ and $alpha_tau = 2$ for complex $sigma$.
Considering the same cube $Phi$ as above, I was able to see that $vol_{canoncial}(f^{-1}(Phi)) = 2^s = 2^s vol_{Lebesgue}(Phi)$. But I am not sure how to pass this to general measurable set $X subset V$.
measure-theory algebraic-number-theory
asked Jan 2 at 22:58
libofmathlibofmath
506
$endgroup$
add a comment |
$begingroup$
I would like to understand what Neukirch means when he writes down
$$ vol_{canonical}(X) = 2^s vol_{Lebesgue}(f(X)) $$
(Neukirch, Algebraic Number Theory. Pg 31)
I will write down the details at the end of the post.
I think it suffices to understand the following situation:
Let $V$ be a Euclidean vector space.
This means that $V$ is a finite-dimensional real vector space with symmetric, positive-definite bilinear form $langle ~,~ rangle : V times V to mathbb{R}$. Let $mu$ be the Haar measure of a Euclidean vector space (such that the cube spanned by the orthonormal basis has measure 1).
Let $e_i in mathbb{R}^n$ be the vector with $1$ on $i$th coordinate and $0$ everywhere else. Let $Phi$ be the cube spanned by $e_i$. Then $mu_{Leb}(Phi)=1$ where $mu_{Leb}$ is the Lebesgue measure on $mathbb{R}^n$.
Question: Let $f: V to mathbb{R}^n$ be a linear isomorphism such that $$ mu(f^{-1}(Phi)) = c $$ or equivalently $$ mu(f^{-1}(Phi))= c mu_{Leb}(Phi) $$ Then for any measurable set $X subset V$, $$ mu(X) = c mu_{Leb}(f(X)) $$
Details: Let $K/mathbb{Q}$ be a number field with $[K:mathbb{Q}]=n$. Let $K_mathbb{R}$ be the set of $n$-tuple of $mathbb{C}$ indexed by $Hom(mathbb{Q}, mathbb{C})$ with the following properties. If $rho_1, ldots, rho_r$ are real embeddings of $K$ and $tau_1, overline{tau_1}, ldots, tau_s, overline{tau_s}$ are the complex embeddings, then elements $(z_sigma)$ in $K_mathbb{R}$ is of the form
$$ z_rho in mathbb{R} text{ and } z_overline{tau} = overline{z_tau} $$
where $sigma$ varies over $Hom(K,mathbb{C})$. $K_mathbb{R}$ becomes a finite-dimensional real vector space, so it has a Haar measure $vol_{canonical}$. Let $f: K_mathbb{R} to mathbb{R}^n$ be a linear isomorphism defined by $(z_sigma) mapsto (x_sigma)$ where
$$ x_rho = z_rho text{ and } x_tau = Re(z_tau), x_overline{tau} = Im(z_tau) $$
$K_mathbb{R}$ has a symmetric, positive-definite bilinear form
$$ langle z, w rangle = sum_sigma z_sigma overline{w_sigma} $$
which transfers to $mathbb{R}^n$ by
$$ (x,y) = sum_sigma alpha_sigma x_sigma y_sigma $$
where $alpha_rho = 1$ for real $rho$ and $alpha_tau = 2$ for complex $sigma$.
Considering the same cube $Phi$ as above, I was able to see that $vol_{canoncial}(f^{-1}(Phi)) = 2^s = 2^s vol_{Lebesgue}(Phi)$. But I am not sure how to pass this to general measurable set $X subset V$.
measure-theory algebraic-number-theory
asked Jan 2 at 22:58
libofmathlibofmath
506
$endgroup$
$begingroup$
$d mu_{leb}(x) = dmu(A x)$ where $A$ is linear and $mu_{leb},mu$ are Haar measures so $d mu_{leb}(x) = a dmu( x)$
$endgroup$
– reuns
Jan 3 at 4:17
add a comment |
$begingroup$
I would like to understand what Neukirch means when he writes down
$$ vol_{canonical}(X) = 2^s vol_{Lebesgue}(f(X)) $$
(Neukirch, Algebraic Number Theory. Pg 31)
I will write down the details at the end of the post.
I think it suffices to understand the following situation:
Let $V$ be a Euclidean vector space.
This means that $V$ is a finite-dimensional real vector space with symmetric, positive-definite bilinear form $langle ~,~ rangle : V times V to mathbb{R}$. Let $mu$ be the Haar measure of a Euclidean vector space (such that the cube spanned by the orthonormal basis has measure 1).
Let $e_i in mathbb{R}^n$ be the vector with $1$ on $i$th coordinate and $0$ everywhere else. Let $Phi$ be the cube spanned by $e_i$. Then $mu_{Leb}(Phi)=1$ where $mu_{Leb}$ is the Lebesgue measure on $mathbb{R}^n$.
Question: Let $f: V to mathbb{R}^n$ be a linear isomorphism such that $$ mu(f^{-1}(Phi)) = c $$ or equivalently $$ mu(f^{-1}(Phi))= c mu_{Leb}(Phi) $$ Then for any measurable set $X subset V$, $$ mu(X) = c mu_{Leb}(f(X)) $$
Details: Let $K/mathbb{Q}$ be a number field with $[K:mathbb{Q}]=n$. Let $K_mathbb{R}$ be the set of $n$-tuple of $mathbb{C}$ indexed by $Hom(mathbb{Q}, mathbb{C})$ with the following properties. If $rho_1, ldots, rho_r$ are real embeddings of $K$ and $tau_1, overline{tau_1}, ldots, tau_s, overline{tau_s}$ are the complex embeddings, then elements $(z_sigma)$ in $K_mathbb{R}$ is of the form
$$ z_rho in mathbb{R} text{ and } z_overline{tau} = overline{z_tau} $$
where $sigma$ varies over $Hom(K,mathbb{C})$. $K_mathbb{R}$ becomes a finite-dimensional real vector space, so it has a Haar measure $vol_{canonical}$. Let $f: K_mathbb{R} to mathbb{R}^n$ be a linear isomorphism defined by $(z_sigma) mapsto (x_sigma)$ where
$$ x_rho = z_rho text{ and } x_tau = Re(z_tau), x_overline{tau} = Im(z_tau) $$
$K_mathbb{R}$ has a symmetric, positive-definite bilinear form
$$ langle z, w rangle = sum_sigma z_sigma overline{w_sigma} $$
which transfers to $mathbb{R}^n$ by
$$ (x,y) = sum_sigma alpha_sigma x_sigma y_sigma $$
where $alpha_rho = 1$ for real $rho$ and $alpha_tau = 2$ for complex $sigma$.
Considering the same cube $Phi$ as above, I was able to see that $vol_{canoncial}(f^{-1}(Phi)) = 2^s = 2^s vol_{Lebesgue}(Phi)$. But I am not sure how to pass this to general measurable set $X subset V$.
measure-theory algebraic-number-theory
asked Jan 2 at 22:58
libofmathlibofmath
506
$endgroup$
I would like to understand what Neukirch means when he writes down
$$ vol_{canonical}(X) = 2^s vol_{Lebesgue}(f(X)) $$
(Neukirch, Algebraic Number Theory. Pg 31)
I will write down the details at the end of the post.
I think it suffices to understand the following situation:
Let $V$ be a Euclidean vector space.
This means that $V$ is a finite-dimensional real vector space with symmetric, positive-definite bilinear form $langle ~,~ rangle : V times V to mathbb{R}$. Let $mu$ be the Haar measure of a Euclidean vector space (such that the cube spanned by the orthonormal basis has measure 1).
Let $e_i in mathbb{R}^n$ be the vector with $1$ on $i$th coordinate and $0$ everywhere else. Let $Phi$ be the cube spanned by $e_i$. Then $mu_{Leb}(Phi)=1$ where $mu_{Leb}$ is the Lebesgue measure on $mathbb{R}^n$.
Question: Let $f: V to mathbb{R}^n$ be a linear isomorphism such that $$ mu(f^{-1}(Phi)) = c $$ or equivalently $$ mu(f^{-1}(Phi))= c mu_{Leb}(Phi) $$ Then for any measurable set $X subset V$, $$ mu(X) = c mu_{Leb}(f(X)) $$
Details: Let $K/mathbb{Q}$ be a number field with $[K:mathbb{Q}]=n$. Let $K_mathbb{R}$ be the set of $n$-tuple of $mathbb{C}$ indexed by $Hom(mathbb{Q}, mathbb{C})$ with the following properties. If $rho_1, ldots, rho_r$ are real embeddings of $K$ and $tau_1, overline{tau_1}, ldots, tau_s, overline{tau_s}$ are the complex embeddings, then elements $(z_sigma)$ in $K_mathbb{R}$ is of the form
$$ z_rho in mathbb{R} text{ and } z_overline{tau} = overline{z_tau} $$
where $sigma$ varies over $Hom(K,mathbb{C})$. $K_mathbb{R}$ becomes a finite-dimensional real vector space, so it has a Haar measure $vol_{canonical}$. Let $f: K_mathbb{R} to mathbb{R}^n$ be a linear isomorphism defined by $(z_sigma) mapsto (x_sigma)$ where
$$ x_rho = z_rho text{ and } x_tau = Re(z_tau), x_overline{tau} = Im(z_tau) $$
$K_mathbb{R}$ has a symmetric, positive-definite bilinear form
$$ langle z, w rangle = sum_sigma z_sigma overline{w_sigma} $$
which transfers to $mathbb{R}^n$ by
$$ (x,y) = sum_sigma alpha_sigma x_sigma y_sigma $$
where $alpha_rho = 1$ for real $rho$ and $alpha_tau = 2$ for complex $sigma$.
Considering the same cube $Phi$ as above, I was able to see that $vol_{canoncial}(f^{-1}(Phi)) = 2^s = 2^s vol_{Lebesgue}(Phi)$. But I am not sure how to pass this to general measurable set $X subset V$.
measure-theory algebraic-number-theory
measure-theory algebraic-number-theory
asked Jan 2 at 22:58
libofmathlibofmath
506
asked Jan 2 at 22:58
libofmathlibofmath
506
asked Jan 2 at 22:58
libofmathlibofmath
506
asked Jan 2 at 22:58
libofmathlibofmath
506
asked Jan 2 at 22:58
libofmathlibofmath
506
506
$begingroup$
$d mu_{leb}(x) = dmu(A x)$ where $A$ is linear and $mu_{leb},mu$ are Haar measures so $d mu_{leb}(x) = a dmu( x)$
$endgroup$
– reuns
Jan 3 at 4:17
add a comment |
$begingroup$
$d mu_{leb}(x) = dmu(A x)$ where $A$ is linear and $mu_{leb},mu$ are Haar measures so $d mu_{leb}(x) = a dmu( x)$
$endgroup$
– reuns
Jan 3 at 4:17
$begingroup$
$d mu_{leb}(x) = dmu(A x)$ where $A$ is linear and $mu_{leb},mu$ are Haar measures so $d mu_{leb}(x) = a dmu( x)$
$endgroup$
– reuns
Jan 3 at 4:17
$begingroup$
$d mu_{leb}(x) = dmu(A x)$ where $A$ is linear and $mu_{leb},mu$ are Haar measures so $d mu_{leb}(x) = a dmu( x)$
$endgroup$
– reuns
Jan 3 at 4:17
add a comment |
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$begingroup$
$d mu_{leb}(x) = dmu(A x)$ where $A$ is linear and $mu_{leb},mu$ are Haar measures so $d mu_{leb}(x) = a dmu( x)$
$endgroup$
– reuns
Jan 3 at 4:17