A set E Rn is said to be of measure zero if and only if given 0 there is a sequence of rectangles R1, R2, which covers E such that k 1 Rk a) Prove that if E Rn is of volume zero, then E is of measure zero b) Prove that if E Rn is at most countable, then E is of measure zero c) Prove that there is a...

a If E is of volume zero then by definition there is a finite collection of rectang ...

A set E Rn is said to be of measure zero if and only if given

Question:

A set E ⊂ Rn is said to be of measure zero if and only if given ε > 0 there is a sequence of rectangles R1, R2,... which covers E such that ∑∞k=1 |Rk| < ε
a) Prove that if E ⊂ Rn is of volume zero, then E is of measure zero.
b) Prove that if E ⊂ Rn is at most countable, then E is of measure zero.
c) Prove that there is a set E ⊂ R2 of measure zero which does not have zero area and, in fact, is not even a Jordan region.