If A is measurable then the translate A+x is measurable. How will I prove this one?
2007-09-10 18:35:58 · 2 answers · asked by MATHer 2 in Science & Mathematics ➔ Mathematics
Depends on what you're allowed to work with. If you're using the Caratheodory criterion for measurability, then the simplest way, I think, is to show that translation preserves outer measure, then use that and the fact that translation commutes with intersection and complement (i.e. the translate of the intersection is the intersection of the translate, and the translate of the complement is the complement of the translate) to show that if A is measurable, then for an arbitrary set B, λ*(B∩(A+x)) + λ*(B∩∁(A+x)) = λ*((B-x)∩A) + λ*((B-x)∩∁A) = λ*(B-x) = λ*(B), so A+x is measurable.
2007-09-17 17:44:54 · answer #1 · answered by Pascal 7 · 0⤊ 0⤋
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2007-09-11 01:44:44 · answer #2 · answered by Anonymous · 0⤊ 5⤋