Final answer:
To prove the property #4 of Lebesgue's outer measure, we start with the definition of the measure and show that for a subset A of the union of countably many open intervals, the measure is less than or equal to the sum of the lengths of the intervals.
Step-by-step explanation:
Lebesgue's outer measure is a measure of the size or length of a set. It is denoted by m*(A) and is defined as the infimum of the sum of the lengths of intervals that cover the set A. The property #4 states that if A is a subset of a countable collection of open intervals {Iₙ}, then the outer measure of A is less than or equal to the sum of the lengths of the intervals in the collection.
To prove this, we start with the definition of the outer measure. Let A be a subset of the union of open intervals ⋃ₙ=₁ᵢⁿ Iₙ. By definition, m*(A) is the infimum of the sum of lengths of coverings of A by intervals. Since A is a subset of the union of the intervals, we can directly cover A by the same collection of intervals {Iₙ}. Therefore, the sum of the lengths of these intervals is an upper bound for m*(A). Hence, we have m*(A) ≤ ∑ₙ=₁ᵢⁿ ℓ(Iₙ), which proves the property #4.