Strange thing

1. Sep 3, 2008

lark

I realized something weird.
That, suppose you take the rationals in [0,1], call this set $$Q.$$ $$Q$$'s a Borel set, so if $$\mu$$ is Lebesgue measure, $$\mu(Q)=inf(\mu(V), V$$open,$$Q \subset V).$$
$$Q$$ can be covered by open sets of total measure $$\le 1$$ by counting the rationals; cover the first rational by an interval size 1/2, the second by an interval size 1/4 ...
But, $$Q$$ can also be covered by open sets of total measure $$\le 1/2$$ in the same way. Or by an open covering of arbitrarily small total measure ...
It's strange considering that the rationals are dense in the irrationals. Yet you could leave 999/1000 of the irrationals out of the open cover ...
Laura

2. Sep 4, 2008

tiny-tim

Hi Laura!

Why is that strange?

Measure is supposed to be like weighing …

if you tipped all the rationals into a pan and weighed them, you wouldn't expect them to weigh anything, would you?

3. Sep 4, 2008

lark

So you're surrounding each rational in [0,1] by an open interval; the rationals are dense in [0,1]; yet it can be arranged so practically all (say 999999 out of a million) of the irrationals are not covered by one of the open intervals. That's what is weird.
Laura

4. Sep 4, 2008

mathman

This observation (rationals are a set of measure zero) is one way of showing that the set of real numbers between 0 and 1 is uncountable. This is useful for those who don't like Cantor's diagonal proof.