description/proof of that for measure space and \(2\) measurable subsets, measure of union of subsets is sum of measures of subsets minus measure of intersection of subsets
Topics
About: measure space
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that for any measure space and any \(2\) measurable subsets, the measure of the union of the subsets is the sum of the measures of the subsets minus the measure of the intersection of the subsets.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\((M, A, \mu)\): \(\in \{\text{ the measure spaces }\}\)
\(\{a_1, a_2\}\): \(\subseteq A\)
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Statements:
\(\mu (a_1 \cup a_2) = \mu (a_1) + \mu (a_2) - \mu (a_1 \cap a_2)\)
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2: Proof
Whole Strategy: Step 1: see that \(a_1 \cup a_2 = (a_1 \setminus (a_1 \cap a_2)) \cup (a_2 \setminus (a_1 \cap a_2)) \cup (a_1 \cap a_2)\), as the exclusive union; Step 2: see that \(\mu (a_1 \cup a_2) = \mu (a_1) + \mu (a_2) - \mu (a_1 \cap a_2)\), by using the exclusive union.
Step 1:
\(a_1 \cup a_2 = (a_1 \setminus (a_1 \cap a_2)) \cup (a_2 \setminus (a_1 \cap a_2)) \cup (a_1 \cap a_2)\), as the exclusive union, by the proposition that for any \(2\) sets, the union of the sets is the exclusive union of the 1st set minus the intersection of the sets, the 2nd set minus the intersection of the sets, and the intersection of the sets.
Step 2:
So, \(\mu (a_1 \cup a_2) = \mu ((a_1 \setminus (a_1 \cap a_2)) \cup (a_2 \setminus (a_1 \cap a_2)) \cup (a_1 \cap a_2)) = \mu (a_1 \setminus (a_1 \cap a_2)) + \mu (a_2 \setminus (a_1 \cap a_2)) + \mu (a_1 \cap a_2)\).
But \(\mu (a_1 \setminus (a_1 \cap a_2)) = \mu (a_1) - \mu (a_1 \cap a_2)\), because \(a_1 = (a_1 \setminus (a_1 \cap a_2)) \cup (a_1 \cap a_2)\), as the exclusive union; \(\mu (a_2 \setminus (a_1 \cap a_2)) = \mu (a_2) - \mu (a_1 \cap a_2)\), likewise.
So, \(= \mu (a_1) - \mu (a_1 \cap a_2) + \mu (a_2) - \mu (a_1 \cap a_2) + \mu (a_1 \cap a_2) = \mu (a_1) + \mu (a_2) - \mu (a_1 \cap a_2)\).
