- #1
MrGandalf
- 30
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I have a sneaky suspicion that this is a fairly simple task, but I just can't seem to break through this particular proof.
Homework Statement
The (Lebesgue) outer measure of any set [itex]A\subseteq\mathbb{R}[/itex] is:
[itex]m^*(A) = inf Z_A[/itex]
where
[itex]Z_A = \bigg\{\sum_{n=1}^\infty l(I_n)\;:\;I_n\;\text{are intervals},\;A\subseteq\bigcup_{n=1}^\infty I_n\bigg\}[/itex]
My problem is to prove that [itex]m^*[/itex] is monotone, i.e
If [itex]A\subset B[/itex] then [itex]m^*(A) \leq m^*(B)[/itex]
The hints are to show that [itex]Z_B \subset Z_A[/itex] and then use the definition of the infimum to show that the larger set can't have an infimum greater than the smaller set.
The attempt at a solution
If [itex]I_n[/itex] covers B, then it will also cover A.
[itex]A\subset B\subset \bigcup_n I_n[/itex]. Hence [itex]Z_B \subset Z_A[/itex].
And now I am confused! :) I have two questions.
1) I was under the impression that [itex]l(I_n)[/itex] was the length of a certain intervall. But that makes [itex]Z_A[/itex] and [itex]Z_B[/itex] numbers. How can one number be a subset of another number?
2) My other question is regarding the statement about infimums. What does the size of a set have to do with what the infimum can be? For instance.
[itex]A = \{2, 3\}[/itex] and [itex]B = \{1,2,3,4\}[/itex]. Here [itex]A\subset B[/itex] but [itex]\text{inf}\; B < \text{inf}\; A[/itex]
Hopefully someone can shed some light on this!
Homework Statement
The (Lebesgue) outer measure of any set [itex]A\subseteq\mathbb{R}[/itex] is:
[itex]m^*(A) = inf Z_A[/itex]
where
[itex]Z_A = \bigg\{\sum_{n=1}^\infty l(I_n)\;:\;I_n\;\text{are intervals},\;A\subseteq\bigcup_{n=1}^\infty I_n\bigg\}[/itex]
My problem is to prove that [itex]m^*[/itex] is monotone, i.e
If [itex]A\subset B[/itex] then [itex]m^*(A) \leq m^*(B)[/itex]
The hints are to show that [itex]Z_B \subset Z_A[/itex] and then use the definition of the infimum to show that the larger set can't have an infimum greater than the smaller set.
The attempt at a solution
If [itex]I_n[/itex] covers B, then it will also cover A.
[itex]A\subset B\subset \bigcup_n I_n[/itex]. Hence [itex]Z_B \subset Z_A[/itex].
And now I am confused! :) I have two questions.
1) I was under the impression that [itex]l(I_n)[/itex] was the length of a certain intervall. But that makes [itex]Z_A[/itex] and [itex]Z_B[/itex] numbers. How can one number be a subset of another number?
2) My other question is regarding the statement about infimums. What does the size of a set have to do with what the infimum can be? For instance.
[itex]A = \{2, 3\}[/itex] and [itex]B = \{1,2,3,4\}[/itex]. Here [itex]A\subset B[/itex] but [itex]\text{inf}\; B < \text{inf}\; A[/itex]
Hopefully someone can shed some light on this!