# Prove that an outer measure πβ on a metric space (π, π) is a Borel outer measure if and only if πβ is a metric outer measure.

**INSTRUCTIONS TO CANDIDATES**
###### ANSWER ALL QUESTIONS

Group-A

Answer any one question from the following questions (15 marks)

1. Prove that an outer measure π∗ on a metric space (π, π) is a Borel outer measure if and only if π∗ is a metric outer measure.

2. Let π be a measure on a π-algebra π of subset of a set π.

(a) For an arbitrary sequence{πΈn: π ∈ β} in π, we

have π(limn→∞πππ πΈn) ≤ limn→∞ ππππ(πΈn). Give an example where the equality doesn’t hold.

(b)(b) If there exists π΄ ∈ π with π(π΄) < ∞ such that πΈn ⊂ π΄ for all π ∈ β then π(limn→∞ sup πΈn) ≥ limn→∞ π π’π π(πΈn). Give an example where the equality doesn’t hold.

(c) If limn→∞ πΈn exists and if there exists π΄ ∈ π with π(π΄) < ∞

such that πΈn ⊂ π΄ for all π ∈ β then limn→∞ π(πΈn) exists and π(limn→∞ πΈn) = limn→∞ π(πΈn).

Group-B

Answer any one question from the following questions (10 marks)

1. Let (π, π) be a measurable space and {πn: π ∈ β} monotone sequence of extended real valued π-measurable functions on a set π· ∈ π. Prove that limn→∞ πn exists on π· and is π-measurable on π·.

2. Let π be a measure on a π-algebra π of subset of a set π.

If {πΈn: π ∈ β} is a decreasing sequence in π and if π(πΈ) < +∞,

then π(β∞ πΈn) = limn→∞ π(πΈn). Show that equality may fail if

the finiteness condition π(πΈ) < +∞, is dropped.

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