Dilations. Mimic the proof of Theorem 5.8 (i) and show that (t cdot B:={t b: b in

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Dilations. Mimic the proof of Theorem 5.8 (i) and show that $t \cdot B:=\{t b: b \in B\}$ is a Borel set for all $B \in \mathscr{B}\left(\mathbb{R}^{n}ight)$ and $t>0$. Moreover,

\[\lambda^{n}(t \cdot B)=t^{n} \lambda^{n}(B) \quad \forall B \in \mathscr{B}\left(\mathbb{R}^{n}ight), \forall t>0\]

Data from theorem 5.8

$the translation invariance of u and \", we see that (1)= k(1)([0,7)"), X" (1)= k(1) X" ([0,7)"), ([0, 1)") =$

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Measures Integrals And Martingales

ISBN: 9781316620243

2nd Edition

Authors: René L. Schilling

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Question Posted: Jan 22, 2024 03:05 AM

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Dilations. Mimic the proof of Theorem 5.8 (i) and show that (t cdot B:={t b: b in

Question:

(i) The n-dimensional Lebesgue measure X" is invariant under translations, i.e. X" (x+B)=X" (B) VxER", VBEB(R"). (5.3) (ii) Every measure u on (R", B(R")) which is invariant under translations and satisfies K = ([0, 1)")

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First we must make sure that 1B is a Borel set if BE B We consider first rectangles I ab where a b R ...View the full answer

Measures Integrals And Martingales

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