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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) < 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].)

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) < 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?

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What does completeness have to do with your question? You asked about connectedness. –  Thomas Andrews Nov 7 '12 at 17:41
    
Also, you've completely messed up the formatting of that quoted question. –  Thomas Andrews Nov 7 '12 at 17:45
    
The $d$ you originally defined was not a meter. The $d$ defined above is a different meter on a different set, closely related but not the same. (Note, for exmaple, it is not defined on all $\mathcal P(X)$, for example.) –  Thomas Andrews Nov 7 '12 at 18:56
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2 Answers

up vote 1 down vote accepted

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.

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First of all, $d(A,B)$ is not even a metric - it is possible for $d(A,B)=0$ when $A\neq B$. Similarly, $m(A\Delta B)$ is not necessarily finite.

So at minimum you need to add $m(X)<+\infty$ and look at $P(X)$ modulo an equivalence relationship $A \sim B$ if and only if $m(A\Delta B)=0$, or you do not even have a metric.

Even then, it is not necessary that the metric space is connected. If $X$ is finite with $m$ the counting function, $m(A)=|A|$, then $\mathcal P(X)$ is just a discrete space with $2^{|X|}$ elements as a topology under the metric $d$.

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