Question

Show that any map $E \mapsto m(E)$ from Lebesgue measurable sets to elements of $[0,+\infty]$ that obeys the above empty set and countable additivity axioms will also obey the monotonicity and countable subadditivity axioms from Exercise 1.2.3, when restricted to Lebesgue measurable sets of course.

   Show that any map $E \mapsto m(E)$ from Lebesgue measurable sets to elements of $[0,+\infty]$ that obeys the above empty set and countable additivity axioms will also obey the monotonicity and countable subadditivity axioms from Exercise 1.2.3, when restricted to Lebesgue measurable sets of course.

An Introduction To Measure Theory (January 2011 Draft)

Terence Tao 1st Edition

Chapter 1, Problem 12

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Show that any map $E \mapsto m(E)$ from Lebesgue measurable sets to elements of $[0,+\infty]$ that obeys the above empty set and countable additivity axioms will also obey the monotonicity and countable subadditivity axioms from Exercise 1.2.3, when restricted to Lebesgue measurable sets of course.

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