Let be a gauge on a, b and let be a fine partition of a, b (a) Show that there exists a 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 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 P

Question: Let be a gauge on [a, b] and let be a -fine partition of [a, b]. (a) Show that there exists a -fine partition

Let Î´ be a gauge on [a, b] and let be a Î´-fine partition of [a, b].
(a) Show that there exists a Î´-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 Î´-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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