Given rectangle $L subseteq mathbb{R}^n$ st. $v(L) = r$, can we cover $L$ by countably many rectangles $Q_i$...
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Consider a rectangle $L = [a_1, b_1 ] times dots times [a_n, b_n]subseteq mathbb{R}^n$ such that $v(L) = r$ , can we cover $L$ by countably many rectangles $Q_i$ such that $sum_{i=1}^{infty}v(Q_i) < r$ ? Note that $v(L) = (b_1 - a_1)dots (b_n-a_n)$ . Now intuitively I think the answer to this is no, however I'm not really sure how I could prove this, in the sense that if I suppose that I can find such a covering I don't see any way to reach a contradiction.
real-analysis measure-theory
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asked Dec 13 '18 at 5:55
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