w**a
发帖数: 1024 | 1 E在R^n 中LEBESGUE 可测当且仅当给定\epsilon >0,
存在 闭集F属于E使得E－F的外测度小于 \epsilon
上面的是一个定理，我想能不能换个说法
F的内点组成的集合是个开集G，并且G属于F，
那么如果E可测的话，
我们是不是也可以找到一个开集G属于E，使得E－G的外测度<\epsilon，
对吗？充分性成立吗？ | B****n
发帖数: 11290 | 2 It's wrong.
Let E be [0,1]\{all the rational numbers}, then outer measure is 1 and there
is no interior point for E.
【在 w**a 的大作中提到】
: E在R^n 中LEBESGUE 可测当且仅当给定\epsilon >0,
: 存在 闭集F属于E使得E－F的外测度小于 \epsilon
: 上面的是一个定理，我想能不能换个说法
: F的内点组成的集合是个开集G，并且G属于F，
: 那么如果E可测的话，
: 我们是不是也可以找到一个开集G属于E，使得E－G的外测度<\epsilon，
: 对吗？充分性成立吗？
| w**a
发帖数: 1024 | 3 given E = [0,1]\{all the rational numbers}, and \epsilon > 0.
what is the closed set F such that $F \subset E$ and outer measure
(E-F) < \epsilon . many thanks....
there
【在 B****n 的大作中提到】
: It's wrong.
: Let E be [0,1]\{all the rational numbers}, then outer measure is 1 and there
: is no interior point for E.
| w**a
发帖数: 1024 | 4 given E = [0,1]\{all the rational numbers}, and \epsilon > 0.
what is the closed set F such that $F \subset E$ and outer measure
(E-F) < \epsilon . many thanks....
there
【在 B****n 的大作中提到】
: It's wrong.
: Let E be [0,1]\{all the rational numbers}, then outer measure is 1 and there
: is no interior point for E.
| B****n
发帖数: 11290 | 5 An Intersection of closed sets is still a closed set.
So you can construct F by an intersection of a sequence of closed sets
whose measures are \{1-\epsilon/2,...,1-\frac{\epsilon}{2^n},...\}
【在 w**a 的大作中提到】
: given E = [0,1]\{all the rational numbers}, and \epsilon > 0.
: what is the closed set F such that $F \subset E$ and outer measure
: (E-F) < \epsilon . many thanks....
:
: there
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