Question

Let E(1,2) be the set of all numbers in (0,1) such that there decimal
representation does not contain 1 and 2. Prove that E(1,2) is lebesgue
measurable and find the lebesgue measure of E(1,2).
Would you please explain it step by step?

Answer 1

Not my thing :(

Answer 2

I would suggest you try to compute the lebesgue measure of E(1,2) first without being rigorous.
Use the principle of inclusion and exclusion to evaluate the probability that a randomly drawn number in the interval (0,1) does not contain a 1 or 2.

Then, use your computation to give a rigorous proof using the properties of the Lebesgue measure.