Final answer:
The student's query is about the existence of a countable disjoint collection of closed intervals that covers a set of finite outer measure, based on the concept of the Vitali covering theorem in real analysis.
Step-by-step explanation:
The question requires a demonstration that for a set t of finite outer measure covered by a collection of closed, bounded intervals in the sense of Vitali, there exists a countable disjoint subcollection. This is related to the Vitali covering theorem. The process of finding this subcollection typically involves iterating through the original collection F, selecting intervals that are disjoint, and ensuring that the union of these selected intervals still covers the set t.
In order to construct such a collection, we might start with one interval from F and then iteratively choose additional intervals that are disjoint from the ones already selected. This selection process relies on the property of closed intervals that they can be ordered and separated. We then approximate the measure of t by the sum of the measures of these disjoint intervals. As t has a finite measure, the process will converge, and we can cover t up to its outer measure with a countable collection of these intervals.
Through this method, we can address the general strategy, without delving into technical details, to show that such a countable disjoint collection exists which aligns with the Vitali covering theorem concept in real analysis.