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 outer measure is equal to its inner measure. The Lebesgue measure of a set is a measure of its size or length.
To find the Lebesgue measure of E(1,2), we can start by finding its outer measure, also known as the Lebesgue outer measure. The outer measure of a set is defined as the infimum of the sum of the lengths of a countable collection of intervals that cover the set.
Let's break down the problem into steps:
Step 1: Finding the outer measure of E(1,2)
- Consider a single interval (a, b) that covers E(1,2). Since the set E(1,2) contains only numbers whose decimal representation does not contain 1 and 2, it follows that a < 0.1 and b > 0.2.
- The length of the interval (a, b) is b - a. Thus, for any interval (a, b) covering E(1,2), its length is greater than 0.2 - 0.1 = 0.1.
- Hence, the outer measure of E(1,2) is at least 0.1.
Step 2: Finding the inner measure of E(1,2)
- For the inner measure, we need to find a set contained in E(1,2), which has a larger measure than E(1,2).
- To do this, let's consider the set D(1,2) consisting of numbers in the interval (0,1) whose decimal representation does contain 1 and/or 2.
- Clearly, E(1,2) is contained in the complement of D(1,2).
- Now, we know that the Lebesgue measure of the complement of a set is equal to the measure of the entire space minus the measure of the set itself. Thus, the measure of the complement of D(1,2) is equal to 1 - the measure of D(1,2).
Step 3: Finding the measure of D(1,2)
- Let's consider the set A(1) consisting of numbers in the interval (0,1) whose decimal representation contains at least one 1. Its measure can be calculated as follows:
- The numbers in A(1) whose decimal representation start with 1 can be represented as 0.1 + n(0.01), where n is a positive integer.
- Since there are countably infinite such numbers, the measure of this subset is given by ∑(n=1 to ∞) (0.01) = 1.
- Similarly, the measure of the set A(2) consisting of numbers in the interval (0,1) whose decimal representation contains at least one 2 is also 1.
- Now, when we consider the set D(1,2), it consists of numbers that either have at least one 1 or at least one 2 in their decimal representation. Hence, D(1,2) = A(1) union A(2).
- The Lebesgue measure of the union of two sets is given by the sum of their individual measures. Thus, the measure of D(1,2) is 1 + 1 = 2.
Step 4: Finding the inner measure of E(1,2)
- As mentioned earlier, the inner measure of E(1,2) is equal to 1 minus the measure of its complement, which is the measure of D(1,2).
- Therefore, the inner measure of E(1,2) = 1 - 2 = -1.
Step 5: Comparing the outer and inner measures
- The outer measure of E(1,2) is at least 0.1, and the inner measure is -1.
- Since the outer measure is greater than or equal to the inner measure, we can conclude that the set E(1,2) is Lebesgue measurable.
Step 6: Finding the Lebesgue measure of E(1,2)
- The Lebesgue measure of E(1,2) is defined as the difference between the outer measure and the inner measure.
- The Lebesgue measure of E(1,2) = outer measure - inner measure = 0.1 - (-1) = 0.1 + 1 = 1.1.
Therefore, the Lebesgue measure of the set E(1,2) is 1.1.