Question

If E1 and E2 are measurable subsets of [a, b], then prove that m(E1) + m(E2) = m(E1 U E2) + m(E1 ∩ E2)​

Answer 1

Answer:

Show that if E1 and E2 are measurable, then m(E1 ∪ E2) + m(E1 ∩ E2) = m(E1) + m(E2). m(E1 ∪ E2)+ m(E1 ∩ E2) = m(E1 \ E2)+ m(E1 ∩ E2)+ m(E2 \ E1)+ m(E1 ∩ E2) = m(E1)+ m(E2), since E1 is a disjoint union of E1 \ E2 and E1 ∩ E2. (E). m∗(U \ E) > ε0 > m∗(U) − m∗(E).

Answer 2

Answer:

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