Lebesgue Outer Measure of Intersection of Two Sets

Question

How do we find the Lebsgue outer measure of below set

Set : drive[DOT]google.com/file/d/1-ZwjDEF8LO5nLPEMHPgx8oNArUDQi19b/view?usp=drivesdk

I know that both of them have Lebesgue outer measure of O, but not sure how to take the lebesgue outer measure of the whole set.

Answer 1

Problem:

We want the Lebesgue outer measure of the intersection (or union) of two sets AAA and BBB.

We are told that both sets individually have outer measure 000.

Step-by-Step Procedure:

Step 1. Recall the definition of outer measure property

1. If E⊆FE \subseteq FE⊆F, then m∗(E)≤m∗(F)m^*(E) \leq m^*(F)m∗(E)≤m∗(F). (Monotonicity)
2. If E=E1∪E2∪…E = E_1 \cup E_2 \cup \dotsE=E1​∪E2​∪…, then m∗(E)≤m∗(E1)+m∗(E2)+…m^*(E) \leq m^*(E_1) + m^*(E_2) + \dotsm∗(E)≤m∗(E1​)+m∗(E2​)+…. (Countable subadditivity)

Step 2. Apply to the intersection A∩BA \cap BA∩B:

1. Notice that A∩B⊆AA \cap B \subseteq AA∩B⊆A.
2. By monotonicity:
3. m∗(A∩B)≤m∗(A).m^*(A \cap B) \leq m^*(A).m∗(A∩B)≤m∗(A).
4. But we are given m∗(A)=0m^*(A) = 0m∗(A)=0.
5. So,
6. m∗(A∩B)≤0.m^*(A \cap B) \leq 0.m∗(A∩B)≤0.
7. Outer measure cannot be negative, so
8. m∗(A∩B)=0.m^*(A \cap B) = 0.m∗(A∩B)=0.

Step 3. Apply to the union A∪BA \cup BA∪B (if needed):

1. By subadditivity:
2. m∗(A∪B)≤m∗(A)+m∗(B).m^*(A \cup B) \leq m^*(A) + m^*(B).m∗(A∪B)≤m∗(A)+m∗(B).
3. Both are 000, so
4. m∗(A∪B)≤0.m^*(A \cup B) \leq 0.m∗(A∪B)≤0.
5. Again, outer measure is never negative, so
6. m∗(A∪B)=0.m^*(A \cup B) = 0.m∗(A∪B)=0.

Final Answer:

1. The intersection has outer measure 000.
2. The union also has outer measure 000.