Let d be a gauge on I (a, b) and suppose that I does not have a d fine partition (a) Let c (a b) Show that at least one of the intervals a, c and c, b does not have a fine partition (b) Construct a nested sequence (In) of subintervals with the length of In equal to (b a) 2n such that In does not have a fine partition

Question: Let d be a gauge on I: = (a, b) and suppose that I does not have a d-fine partition. (a) Let c : =

Let d be a gauge on I: = (a, b) and suppose that I does not have a d-fine partition.
(a) Let c : = (a + b). Show that at least one of the intervals [a, c] and [c, b] does not have a Î´-fine partition.
(b) Construct a nested sequence (In) of subintervals with the length of In equal to (b - a)/2n such that In does not have a Î´-fine partition.

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