TITLE: Is $\mathcal P(X)$ connected when $(X,\mathcal P(X),m)$ is a measure space and $P(X)$ is equipped with the metric $d(A,B) =m(A\Delta B)$?
QUESTION [6 upvotes]: Is $\mathcal P(X)$ connected when $(X,\mathcal P(X),m)$ is a measure space and $P(X)$ is equipped with the metric $d(A,B) =m(A\Delta B)$? 
Think when we look at the equivalence classes of almost everywhere relation .
What about lebegu measure and X be a subset of Real number.
Also 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) < \infty}. 
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].) 
I just bring above problem to show readers places that d is meter. I need to know if m be lebegu measure and X be a subset of Real line, is {A in P(X) | m(A) is finite} for example when X=[a,b] that a,b are real numbers, a connected space with topology induced with meter d?
Some of you have read this question and here is the last manner, I had written question with lots of ambiguity. Let me to say it again( with no ambiguity ):
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) < \infty}. 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?

REPLY [2 votes]: This addresses your final question: if $X \subset \mathbb{R}$ and $m$ is Lebesgue measure, is $(D,d)$ a connected metric space?
Hint: given $A \subset \mathbb{R}$ measurable, define $C : [0,+\infty] \to D$ by $C(t) = (A \cap (-t,t))^*$.  Show that $C$ is continuous, and that $C(0) = \emptyset^*$, $C(+\infty) = A^*$.  Conclude that $D$ is path connected.