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I think I need Cavalieri: $(X, \mathcal{A}, \mu),(Y, \mathcal{B}, \nu)$ are $\sigma$-finite measure spaces. For $E \in \mathcal{A} \otimes \mathcal{B}$ is $$ \mu \otimes \nu(E)=\int_X \nu\left(E_x\right) \mathrm{d} \mu(x)=\int_Y \mu\left(E^y\right) \mathrm{d} \nu(y)\; . $$ But i dont know how i find the two equal Integrals. And i dont understand the or. $$ \lambda_2(M) = \int_{-1}^{0} (x^2 + y^2) d\lambda_1(x) $$ for $x\le 0$? This is not right.

After the Definition I have now: $$ \lambda_2(M) =\int_{\mathbb{R}} \lambda_1(M_x) d\lambda_1(x) = \int_{\mathbb{R}} \lambda_1(M^y) d\lambda_1(y) $$ But what is $\lambda_1(M_x)$ and $\lambda_1(M^y)$?

Can someone help?

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Daniele Tampieri
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MathJason
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  • Do we agree that region M is the union of the two regions in your graphics ? 2) What is $\lambda_2(M)$ ? Area ? Perimeter ? 3) You present it as a 2D issue ; Is it the case ?
    • – Jean Marie Nov 29 '23 at 17:35
  • Why are your two regions graphed separately windows scaled differently? Presumably, you're interested in some integral calculation on this region in the plane, consisting of two circular sectors. – Sammy Black Nov 29 '23 at 17:42
  • Yes is one. i didnt know how to do it with Wolfram Alpha as one. 2) I think its an Area. But i dont know i have no more Information than this. 3) i think we are in 3D – MathJason Nov 29 '23 at 17:47
  • What is your definition of $\lambda1(x)$ ? Lebesgue measure, but why the "(x)' ? – Jean Marie Nov 29 '23 at 17:57
  • This means that we integrate after x – MathJason Nov 29 '23 at 18:05