Let be a gauge on [a, b] and let be a -fine partition of [a, b].

Let δ be a gauge on [a, b] and let be a δ-fine partition of [a, b].
(a) Show that there exists a δ-fine partition 1 such that (i) no tag belongs to two subintervals in 1, and (ii) S(f; 1) = S(f; ) for any function f on [a, b].
(b) Does there exist a δ-fine partition 2 such that (j) every tag belongs to two subintervals in 2, and (jj) S(f; 2) = S(f; ) for any function f on [a, b]?
(c) Show that there exists a d-fine partition 3 such that (k) every tag is an endpoint of its subinterval, and (kk) S(f; 3) = S(f; ) for any function f on [a, b].

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