Proving Measure Zero for Set A Derived from Real Numbers Set E

[arvindam]

A set E subset of real numbers has measure zero. Set A={x² : x$$\in$$E}. How to prove that set A has measure zero?
(E could be any unbounded subset of R)

 

[micromass]

I would approximate E by open intervals. That is: there exists open intervals $$]a_i,b_i[$$ such that

$$E\subseteq \bigcup]a_i,b_i[$$

and such that

$$\sum{b_i-a_i}<\epsilon$$.

Now square the open intervals to obtain an approximation of A.

 

[arvindam]

But intervals are not bounded. b²_i-a²i does not have a bound when b_--a_i is bounded.

 

[lavinia]

But intervals are not bounded. b²_i-a²i does not have a bound when b_--a_i is bounded.

it is easy for any bounded set of measure zero. Split your set up into countably many of these.