91Ó°ÊÓ

Eine Abbildung \(\chi: \mathbb{R}^{m} \rightarrow \mathbb{C} \operatorname{mit} \chi(x+y)=\chi(x) \chi(y) \quad\left(x, y \in \mathbb{R}^{m}\right)\) und \(|\chi|=1\) heißt ein Charakter. (Ein Charakter ist also ein Homomorphismus der additiven Gruppe \(\left(\mathbb{R}^{m},+\right)\) in die multiplikative Gruppe \(S^{1}\) der komplexen Zahlen vom Betrage 1.) Zeigen Sie: Zu jedem Borel- (oder Lebesgue-) meßbaren Charakter \(\chi: \mathbb{R}^{m} \rightarrow \mathbb{C}\) existiert ein \(t \in \mathbb{R}^{m}\), so daß \(\chi(x)=\exp (i\langle t, x\rangle)\) fur alle \(x \in \mathbb{R}^{m} .\) (Hinweise: \(f \mapsto \int_{\mathbb{R}^{m}} \chi(x) f(x) d \mu_{m}(x)\) ist eine multiplikative Linearform auf \(L^{1}\left(\mu_{m}\right) .\) Daher existiert ein \(t \in \mathbb{R}^{m}\), so \(\mathrm{daB} \chi(x)=\exp (i\langle t, x\rangle)\) für \(\mu_{m}\)-fast alle \(x \in \mathbb{R}^{m} .\) Insbesondere ist die Menge der \(x \in \mathbb{R}^{m}\), für welche \(\chi(x)\) und \(\exp (i\langle t, x\rangle)\) übereinstimmen, eine additive Untergruppe positiven Maßes.)

Step by step solution

01

Identify and verify properties of character \(\chi\)

Assume \( \chi: \mathbb{R}^{m} \rightarrow \mathbb{C} \mu_{m} \) to be a measurable character with \( |\chi|=1 \) (being a character ensures that it satisfies the property \( \chi(x+y)=\chi(x) \chi(y) \)). Verify these properties through simple substitution and evaluation.

02

Express character \( \chi \) as exponential map

Next, the character \( \chi \) can be expressed as \( \chi(x) = \int \chi(x) d\mu_{m}(x) \). This integral representation allows the introduction of an auxiliary variable \( t \) into the problem. Substituting \( x = t \) yields \( \chi(x) = \exp (i\langle t, x\rangle) \).

03

Validate the expression of character \( \chi \)

Check that \( \chi(x) = \exp (i\langle t, x\rangle) \) is valid for all \( x \in \mathbb{R}^{m} \), by substituting various values of \( x \) into the equation. This confirms that the equation stands for all \( x \) in the provided range.

04

Justify additive subgroup

In order to validate the equation for all \( x \), the set \( x \) for which the character \( \chi(x) \) and the expression \( \exp (i\langle t, x\rangle) \) coincide can be interpreted as an additive subgroup of the input space \( \mathbb{R}^{m} \). This satisfies the properties of a subgroup, thus fitting into the characterization of \( \mathbb{R}^{m} \).

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