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Calculus and Beyond Homework Help
Outer Content ( Lebesgue Measure )
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[QUOTE="STEMucator, post: 4491615, member: 426227"] [h2]Homework Statement [/h2] Two parts to this question. a) Suppose ##S## is a bounded set and has an interior point ##Q##, prove that ##C(S) > 0##. ( Hint : Inside any neighbourhood of ##Q##, you can place a [U]square[/U] ). b) For each integer ##n>0##, let ##L_n## be the line segment ##\{ (\frac{1}{n}, y) : 0 ≤ y ≤ \frac{1}{n} ) \}##. Also, let ##S = \bigcup_{n=1}^{∞} L_n##, prove that ##C(S) = 0##. [h2]Homework Equations[/h2] ##C(S)## = Outer content = ##inf \{ \sum A_i \} = inf \{ Area(P) \}##. Where ##A_i## is the area of the sub-rectangle ##R_i##, which comes from a larger rectangle ##R## that encloses ##S## and has been partitioned by ##P##. [h2]The Attempt at a Solution[/h2] a) This part is not too bad I think. Since ##Q## is an interior point of ##S##, ##\exists \delta > 0 \space | \space N_{\delta}(Q) \subseteq S##. Now, place a square, with center ##Q##, inside of ##N_{\delta}(Q)## and let ##A_{S_Q} = Area(Square_Q)##. Suppose that we now enclose ##S## in a rectangle ##R##. For any partition ##P## of ##R##, we can attain another partition ##P'## from the square, which yields the following relationship : ##inf \{ \sum A_i \} ≥ inf \{ \sum A_{i}^{'} \} ≥ A_{S_Q} > 0##. Hence the result is shown, ##C(S) ≥ A_{S_Q} > 0##. b) This part is a bit more difficult. I was thinking to take a square with area ##δ^2## that will contain all the line segments ##L_n## except a finite amount of them. [/QUOTE]
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Outer Content ( Lebesgue Measure )
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