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?
Not my thing :(
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.
To prove that E(1,2) is Lebesgue measurable and find its Lebesgue measure, we can follow the following steps:
Step 1: Define the set E(1,2)
The set 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.
Step 2: Show that E(1,2) is Lebesgue measurable
For a set to be Lebesgue measurable, we need to show that its outer measure is equal to its inner measure. Let's break it down:
Inner Measure: To find the inner measure of E(1,2), denoted as m*(E(1,2)), we need to find the smallest possible open set O that contains E(1,2), and calculate its Lebesgue measure, m(O).
- Every number x in E(1,2) can be represented as a decimal:
x = 0.x1x2x3...
- Let O consist of all numbers whose decimal representation does not contain 1 and 2, except for possibly finitely many digits after the decimal point.
- The Lebesgue measure of O, m(O), is equal to the length of the interval (0, 1), which is 1.
- Since O is the smallest possible open set containing E(1,2), then the inner measure of E(1,2) is m*(E(1,2)) = m(O) = 1.
Outer Measure: To find the outer measure of E(1,2), denoted as m*(E(1,2)), we need to find the largest possible open set O' contained within (0,1), where all numbers in O' have decimal representations that contain 1 and/or 2. This means there must be no numbers in (0,1) that are not in E(1,2).
- The largest possible open set O' that satisfies the given condition is the set of all numbers whose decimal representation contains 1 or 2.
- The Lebesgue measure of O', m(O'), is equal to the length of an interval between 0 and the largest number whose decimal representation contains only the digits 1 and/or 2. In this case, it would be the number 0.2222... (repeating 2s), which is equal to 2/9.
- Since O' is the largest possible open set contained in (0,1) where no numbers in (0,1) are excluded, then the outer measure of E(1,2) is m*(E(1,2)) = m(O') = 2/9.
Since the outer measure m*(E(1,2)) is equal to the inner measure m*(E(1,2)), we can conclude that E(1,2) is Lebesgue measurable.
Step 3: Find the Lebesgue measure of E(1,2)
The Lebesgue measure of E(1,2) is equal to the inner measure or the outer measure of the set, which we have already calculated to be 1.
Therefore, the Lebesgue measure of E(1,2) is 1.