# zero measure

• Mar 26th 2010, 04:30 PM
vinnie100
zero measure
How do you show that any subset of a zero measure set has measure zero?
• Mar 26th 2010, 04:40 PM
Plato
Quote:

Originally Posted by vinnie100
How do you show that any subset of a zero measure set has measure zero?

Is the measure monotone?
• Mar 26th 2010, 05:08 PM
southprkfan1
Quote:

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
• Mar 26th 2010, 06:12 PM
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)
• Mar 26th 2010, 11:32 PM
Drexel28
Quote:

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?