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I have a task where i should check if subsets are Lebesgue measurable and if the Lebesgue measure is finite. One of the subsets is: $$B:=\bigcup\limits_{k=1}^{\infty} \left(\frac{k-1}{k+1},\frac{k}{k+2}\right]\times \left(\frac{1}{k},k\right]\subset\mathbb{R}^2$$ I don't get how i can apply the definition for a Lebesgue measure (https://en.wikipedia.org/wiki/Lebesgue_measure) here. Can someone help me?

Peirot
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    Since the sets in the union are disjoint, the measure of $B$ is the sum of the measures of the rectangle-like sets, if that series converges. – aschepler Jun 28 '22 at 21:52
  • Does the series looks like this than? $$\sum\left(\frac{k}{k+2}-\frac{k-1}{k+1}\right)*\left(k-\frac{1}{k}\right)$$ Since "Any Cartesian product of intervals [a, b] and [c, d] is Lebesgue-measurable, and its Lebesgue measure is (b − a)(d − c), the area of the corresponding rectangle." – Peirot Jun 29 '22 at 11:39

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