look at Problem14.12 of chapter3 of "Aliprantis-Burkinshaw-Principles of real analysis-3ed.1998" ; 12. Let A be the collection of all measurable subsets of X of finite measure. That is, A = {B in X: m(A) < oo}. a. Show that A is a semiring. b. Define a relation ~ on A by B ~ C if m(B \Delta C) = 0. Show that ~ is an equivalence relation on A. c. Let D denote the set of ail equivalence classes of A. For B in A let B* denote the equivalence class of B in D. Now for B*, C* in D define d(B*, C*) = m(B \Delta C). Show that d is well defined and that (D, d) is a complete metric space. (For this part, see also Exercise [3] of Section [31].)

Now let m be lebesgu measure on Real numbers and X be a subset of Real line, is (D,d) a connected space with topology induced with meter d?