Lebesgue measure of some spaces...

Hi,

Again, I've seen these questions in some other forums, and I wonder how it is solved...

**1.**

Let A be the set of x in [0,1] such that the decimal expansion of $\displaystyle x^2$ does not contain any 9.

What is the Lebesgue measure of A ?

*My thoughts on it :*

The easiest answer would be 1, by proving that A is a countable set. But I don't think it's true, because in particular, 0.95² ~ 0.9

So, A is at least of Lebesgue measure 0.05, isn't it ? (Surprised)

**2.**

Let $\displaystyle (X,B,\mu)$ be a measured space and $\displaystyle f ~:~ X \to \mathbb{C}$ a measurable function.

Say why $\displaystyle B=\{x \in X ~:~ |f(x)|\geq \alpha\}$ belongs to B.

We assume that $\displaystyle \mu(B)\neq 0$

Show that there exists $\displaystyle \alpha>0$ such that $\displaystyle \mu\left(\{x \in X ~:~ |f(x)|\geq \alpha\}\right)>0$

That's the exact way it was asked, and I don't really know if it is normal that we have to prove that B belongs to B o.O Maybe it's 2 different sets. So let $\displaystyle C=\{x \in X ~:~ |f(x)|\geq \alpha\}$

Anyway, I don't see how to find D such that $\displaystyle f^{-1}(D)=C$

I guess I'm already rusty with measure theory (Worried)