Real Analysis: Prove that the interval [0,1] is not a zero set

[datenshinoai]

Homework Statement

Prove that the interval [0,1] is not a zero set.

2. The attempt at a solution

Assume for contradiction that the interval [0,1] = Z is a zero set. This mean that given epsilon greater than 0, there is a countable coverage of Z by open intervals (ai, bi) (___I don't know what those intervals should be...___) such that the summation of bi - ai is less than epsilon.

But since Z is uncountable, there cannot be a countable coverable of Z by open intervals. Thus Z is not a zero set.

 

[morphism]

Unfortunately that's not very convincing. So what if Z is uncountable? It can still be covered by countably many open intervals. (And in fact there are uncountable zero sets. Can you think of one?) The point is that it can't be covered by countably many intervals whose total length is small -- after all, the length of [0,1] is 1!

 

[datenshinoai]

Perhaps my question really is, what is the definition of a zero set? The book doesn't have a clear definition.

 

[Hurkyl]

The particulars depend on context. Usually, a zero set is the set of solutions to an equation f(x)=0. But the definition can vary, by restricting which kinds of functions you can use, whether the this criterion is applied locally or globally, and other various things. Surely your book gives an explicit definition somewhere?

 

[morphism]

The particulars depend on context. Usually, a zero set is the set of solutions to an equation f(x)=0. But the definition can vary, by restricting which kinds of functions you can use, whether the this criterion is applied locally or globally, and other various things. Surely your book gives an explicit definition somewhere?

I think in this case a zero set means a subset of R of (Lebesgue) measure zero.

datenshinoai, think of this in terms of 'length.' A zero set has zero 'length.' To be precise, a set is a zero set iff you can cover it with countably many intervals of arbitrarily small total length. (At least this is what your definition appears to be.)

 

[Dick]

Perhaps my question really is, what is the definition of a zero set? The book doesn't have a clear definition.

You gave the definition of a zero set in your problem statement. It's a set that can be covered by a union of open intervals with arbitrarily small total length. The set of rational numbers in [0,1] can be covered in such a way. Why not the set of all REAL numbers in [0,1]?