How to calculate Lebesgue integral on a difference of balls?
Such as:
$\int_{B(0,2) \setminus {\overline{B}(0,1)}}$
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$\overline{B}(0,1)\subset B(0,2)$ thus $$ \int_{B(0,2)\setminus\overline{B}(0,1)}d\mu=\int_{B(0,2)}d\mu-\int_{\overline{B}(0,1)}d\mu $$
Tuvasbien
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Good. What if the latter integral happens to be $\infty$? – mavavilj Oct 15 '20 at 19:19
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The Lebesgue integral of a ball cannot be $+\infty$. – Tuvasbien Oct 15 '20 at 19:23
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Consider $\lim_{i \rightarrow \infty} \int_{\overline{B}(0,1)} \log(1+\frac{1}{|x|^i})$? The inside is infinity. – mavavilj Oct 15 '20 at 19:25
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Oh, I thought you were considering $\int_{B}1d\mu$. In this case the previous identity is true only if the two integrals on the right side are finite. – Tuvasbien Oct 15 '20 at 19:28