I am trying to learn about Lebesgue measure. One of the questions I couldn't solve is this;(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Trying to find the measure of a set.

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