Question:
For an interval of the form (-infinity, b) or (-infinity, b], show that m*(I) = l(I), where m*(I) and l(I) denote the Lebesgue Outer Measure of I and the length of interval I, respectively.
My approch towards the question:
Let I be an interval of the form (-infinity, b) or (-infinity, b]
Let k be any positive number.
Then, [b-2k, b-k] is a subset of I.
Then, m*([b-2k, b-k]) <= m*(I)
===> (b-k) - (b-2k) <= m*(I)
===> k<= m*(I)
Since this is true for any positive number k, we must have,
m*(I) = infinity = l(I)
My doubts are whether,
(i) k should be any positive number or k should be any positive real number?
(ii) Is this a concrete proof for the required result?
Your help is highly appreciated!
Your approach is not quite correct because it focuses only on showing that the Lebesgue outer measure is infinite. In this case, both the Lebesgue outer measure and the length of the interval are indeed infinite, but the proof needs to be more general to show that the Lebesgue outer measure and interval length are equal for intervals of the form (-infinity, b) or (-infinity, b].
To show that m*(I) = l(I) for intervals of this form, consider the following approach:
1) First, let's show that l(I) ≤ m*(I).
For any interval I of the form (-infinity, b) or (-infinity, b], the length l(I) = b - (-infinity) = infinity.
Given any ε > 0, consider the countable collection of open intervals {O_n} such that O_n = (b - 2^n - ε/2^(n+1), b - 2^n) for n ∈ ℕ.
You can see that I ⊆ ⋃O_n, and that the sum of the lengths of the O_n intervals is less than ε (each interval has length ε/2^n, and summing them over n ∈ ℕ gives ε by the geometric series formula).
Since the Lebesgue outer measure m*(I) is defined to be the infimum of the sums of the lengths of countable collections of open intervals covering I, we have:
l(I) = infinity ≤ m*(I) + ε
Since ε > 0 is arbitrary, l(I) ≤ m*(I).
2) Now, let's show that m*(I) ≤ l(I).
Let {O_n} be any countable collection of open intervals such that I ⊆ ⋃O_n.
Notice that for each n, there exists a positive integer N such that b - 2^N ∈ O_n.
Thus, the sum of the lengths of the O_n intervals must be at least infinity (as we can find arbitrarily large N for which b - 2^N ∈ O_n) which means that:
m*(I) ≤ l(I) = infinity
Since we have shown that l(I) ≤ m*(I) and m*(I) ≤ l(I), we can conclude that m*(I) = l(I) for intervals of the form (-infinity, b) or (-infinity, b].
To address your specific doubts:
(i) k can be any positive real number. However, in the general proof provided above, we used positive integers for simplicity.
(ii) The provided general proof above is a more concrete proof for showing that the Lebesgue outer measure and interval length are equal for intervals of this form.