Trying to find the measure of a set.

[Jamesandthegi]

I am trying to learn about Lebesgue measure. One of the questions I couldn't solve is this;

Show that the following sets are Lebesgue measurable and determine their measure

A = {x in [0,1) : the nth digit in the decimal expansion is equal to 7}

B = {x in [0,1) : all but finitely many digits in the decimal expansion are equal to 7}

Now, the book defines a set E to be Lebesgue measurable if E = A U B, where A is in the Borel $\sigma$-algebra and B is a null set (outer measure 0), but I don't see where that helps here. Any hints?

 

[Jamesandthegi]

Ok, I figured out A, but I'm not sure on B. Any help?

 

[mathman]

The measure is 0. The general idea is that for any n if n digits are 7, then the measure is (1/10)ⁿ. So let n -> ∞

 

[Jamesandthegi]

Yes, but that doesn't prove that it's measurable. To prove its so, you have to write it as A U B, A borel and B null. Then the measure part I get.

 

[mathman]

If its null it is measurable. You can always union it with the empty set of you insist on having a Borel set to union with.

 

[wofsy]

For a fixed finite set of decimal place not equal to seven the number of points is finite. So for instance the number of points if only the first 2 place are not 7 is 100. The number of finite subsets of a countable set is countable, I think. So the union of all of these finite sets is countable and thus Borel measurable zero.