Any one?
Helle Everyone,
In one book, the Lebesgue measure is said to possess the following properties (among others):
1) the measure of any measurable set can be approximated from above by open sets; that is, for any measurable M
is open
2) the measure of any measurable set can be approximated from below by compact sets; that is, for any measurable M
is compact
Can any one provide a very simple example of this property? Like in ?
Are you after something like this?
Bolzano?Weierstrass theorem - Wikipedia, the free encyclopedia
If all you want is an example, take [0, 1]. It can be "approximated from above" by the open sets (-1/n, (n+1)/n). Each of those has measure (n+1)/n+ 1/n= 1+ 2/n which goes to 1, the measure of [0, 1], as n goes to infinity.
Again, use [0, 1] which can be "approximated from below" by the compact sets [1/n, (n-1)/n]2) the measure of any measurable set can be approximated from below by compact sets; that is, for any measurable M
is compact
Can any one provide a very simple example of this property? Like in ?