Thread: zero measure

How do you show that any subset of a zero measure set has measure zero?

Originally Posted by vinnie100

How do you show that any subset of a zero measure set has measure zero?

Is the measure monotone?

Originally Posted by vinnie100

How do you show that any subset of a zero measure set has measure zero?

Well, if we're talking Lebesgue measure...

Let Z be a zero set and A be a subset of Z

we know:

m*(A) >= 0 (by definition)

and m*(A) <= m*(Z) = 0, this follows by monotonicity

so 0 <= m*(A) <= 0

and m*(A) = 0

We defined a set to have measure zero if for any epsilon>0, there is (at most countable) collection of intervals that cover A and whose total length is less than epsilon.

Hint: For A_n, choose a cover with epsilon_n = epsilon/(2^n)

Originally Posted by vinnie100

We defined a set to have measure zero if for any epsilon>0, there is (at most countable) collection of intervals that cover A and whose total length is less than epsilon.

Hint: For A_n, choose a cover with epsilon_n = epsilon/(2^n)

Why wouldn't the cover for the superset work for the subset?