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).
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To prove that E(1,2) is Lebesgue measurable, we need to show that its Lebesgue measure can be determined. The Lebesgue measure of a set is a measure of its size or volume in the Lebesgue space.
To find the Lebesgue measure of E(1,2), we can use the following steps:
Step 1: Define the set E(1,2) precisely.
E(1,2) is defined as the set of all numbers in the interval (0,1) whose decimal representation does not contain the digits 1 and 2. In other words, E(1,2) consists of all real numbers between 0 and 1 (exclusive) whose decimal expansion only contains digits other than 1 and 2.
Step 2: Simplify the problem by considering the complement of E(1,2).
To find the Lebesgue measure of E(1,2), it is often easier to find the measure of the complement of E(1,2), denoted by E'(1,2). The complement of E(1,2) consists of all numbers in (0,1) whose decimal expansion includes the digits 1 or 2.
Step 3: Expressing the complement E'(1,2) as a countable union of disjoint intervals.
Each number in E'(1,2) can be represented as an infinite sequence of digits: x = 0.x1x2x3... In this sequence, there will exist at least one digit that is 1 or 2 since we are considering the complement of E(1,2).
Let's consider an interval [1,2) that contains all numbers whose first digit is 1. This interval represents a subset of E'(1,2) since any number with a first digit of 1 cannot be in E(1,2). The measure of this interval can be calculated as 1-0 = 1.
Similarly, we can consider intervals [0.1, 0.2), [0.01, 0.02), [0.001, 0.002), and so on, each containing numbers whose second, third, fourth, and so on digit is 1. The measure of each of these intervals can also be calculated as 1-0 = 1.
Since there are infinitely many intervals, we can represent E'(1,2) as a countable union of these disjoint intervals:
E'(1,2) = [1,2) ∪ [0.1, 0.2) ∪ [0.01, 0.02) ∪ [0.001, 0.002) ∪ ...
Step 4: Evaluate the Lebesgue measure of E'(1,2).
The Lebesgue measure of a countable union of disjoint intervals can be determined by adding their individual measures. Since each interval has a measure of 1, the Lebesgue measure of the complement E'(1,2) is the sum of the measures of these intervals:
m(E'(1,2)) = m([1,2)) + m([0.1, 0.2)) + m([0.01, 0.02)) + m([0.001, 0.002)) + ...
Since each interval has a measure of 1, the sum of the measures of these intervals is infinite. Therefore, m(E'(1,2)) = ∞.
Step 5: Use the definition of Lebesgue measure to find the measure of E(1,2).
By definition, the Lebesgue measure of a set E is given by:
m(E) = m(U) - m(E'),
where m(U) is the measure of the entire space, which in this case is (0,1), and m(E') is the measure of the complement of E.
Substituting the values, we have:
m(E(1,2)) = m((0,1)) - m(E'(1,2)) = 1 - ∞
Since infinity is not a valid measure, the Lebesgue measure of E(1,2) is undefined.
In conclusion, E(1,2) is Lebesgue measurable, but its Lebesgue measure is undefined.