Let E(1,2) be the set of all numbers in (0,1) such that their 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).
To prove that the set E(1,2) is Lebesgue measurable, we need to show that its Lebesgue outer measure is equal to its Lebesgue measure.
First, let's define the set E(1,2) more precisely. E(1,2) is the set of all numbers in the interval (0,1) whose decimal representation does not contain 1 and 2. In other words, E(1,2) consists of all numbers in (0,1) that can be written as a decimal between 0.02 and 1.00, with no digits 1 or 2 in their decimal representation.
To find the Lebesgue measure of E(1,2), we need to find its Lebesgue outer measure first. The Lebesgue outer measure of a set A, denoted as m*(A), is defined as the infimum of the sums of the lengths of intervals covering A. In other words, it is the smallest possible sum of lengths of intervals that cover the set A.
To calculate the Lebesgue outer measure of E(1,2), we can start by covering E(1,2) with a sequence of intervals whose sum of lengths is less than any given positive number ε. For example, we can construct a sequence of intervals that covers E(1,2) in the following way:
1. Divide the interval (0,1) into ten equal subintervals: (0,0.1), (0.1,0.2), ..., (0.9,1).
2. Exclude the subinterval (0,0.02), as it contains numbers that do not belong to E(1,2).
3. Exclude the nine subintervals (0.12,0.13), (0.21,0.22), ..., (0.92,0.93), as they contain numbers that have the digits 1 or 2 in their decimal representation.
4. We are now left with eight subintervals: (0.02,0.12), (0.13,0.21), (0.22,0.3), ..., (0.9,1).
5. We can further divide each of these eight subintervals into ten equal subintervals.
6. Repeat the process described in step 3 for each of the resulting subintervals.
7. Iterate this process indefinitely.
By taking the union of these intervals at each step, we can construct a sequence of nested intervals that cover E(1,2). The sum of the lengths of these intervals decrease as we proceed, and it can be shown that the sum of the lengths of these intervals converges to 1. Therefore, we can say that the Lebesgue outer measure of E(1,2) is 1.
Since the Lebesgue outer measure of E(1,2) is equal to 1, which is the length of the interval (0,1), we can conclude that E(1,2) is Lebesgue measurable.
In summary, to prove that E(1,2) is Lebesgue measurable, we showed that its Lebesgue outer measure is equal to its Lebesgue measure, which is 1 in this case.