Assume A and B are bounded subsets of [0, 1] and A is contained in B. Prove that that mu*(A)<=mu*(B).
Here are the definitions I know (sorry I do not know how to write in the symbols):
mu* of a bounded set S is the outer (Lesbegue) measure defined as glb{m(G): G is open and bounded and S is contained in G} For an open G = (infinite)U Ij, whereIj are pairwise disjoint intervals that cover S, m(G) = sum of the m(Ij).
So I know I can define a covering of open intervals for B since it is bounded, but can I define a covering of disjoint open intervals?
If not, then I am not sure I have any other ideas of how to approach this. Any suggestions to get me going would be appreciated.
Here are the definitions I know (sorry I do not know how to write in the symbols):
mu* of a bounded set S is the outer (Lesbegue) measure defined as glb{m(G): G is open and bounded and S is contained in G} For an open G = (infinite)U Ij, whereIj are pairwise disjoint intervals that cover S, m(G) = sum of the m(Ij).
So I know I can define a covering of open intervals for B since it is bounded, but can I define a covering of disjoint open intervals?
If not, then I am not sure I have any other ideas of how to approach this. Any suggestions to get me going would be appreciated.