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Prove that an outer measure πœ‡βˆ— on a metric space (𝑋, 𝑑) is a Borel outer measure if and only if πœ‡βˆ— 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œ‡∗ on a metric space (𝑋, 𝑑) is a Borel outer measure if and only if πœ‡∗ is a metric outer measure.

 

2. Let πœ‡ be a measure on a 𝜎-algebra π’œ of subset of a set 𝑋.

(a) For an arbitrary sequence{𝐸n: 𝑛 ∈ β„•} in π’œ, we

have πœ‡(limn→∞𝑖𝑛𝑓 𝐸n) ≤ limn→∞ π‘–π‘›π‘“πœ‡(𝐸n). Give an example where the equality doesn’t hold.

(b)(b) If there exists 𝐴 ∈ π’œ with πœ‡(𝐴) < ∞ such that 𝐸n ⊂ 𝐴 for all 𝑛 ∈ β„• then πœ‡(limn→∞ sup 𝐸n) ≥ limn→∞ 𝑠𝑒𝑝 πœ‡(𝐸n). Give an example where the equality doesn’t hold.

(c) If limn→∞ 𝐸n exists and if there exists 𝐴 ∈ π’œ with πœ‡(𝐴) < ∞

such that 𝐸n ⊂ 𝐴 for all 𝑛 ∈ β„• then limn→∞ πœ‡(𝐸n) exists and πœ‡(limn→∞ 𝐸n) = limn→∞ πœ‡(𝐸n).

 

 

 

 

 

 

 

 

Group-B

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

1. Let (𝑋, π’œ) be a measurable space and {𝑓n: 𝑛 ∈ β„•} monotone sequence of extended real valued π’œ-measurable functions on a set 𝐷 ∈ π’œ. Prove that limn→∞ 𝑓n exists on 𝐷 and is π’œ-measurable on 𝐷.

2. Let πœ‡ be a measure on a 𝜎-algebra π’œ of subset of a set 𝑋.

If {𝐸n: 𝑛 ∈ β„•} is a decreasing sequence in π’œ and if πœ‡(𝐸) < +∞,

then πœ‡(β‹‚∞ 𝐸n) = limn→∞ πœ‡(𝐸n). Show that equality may fail if

the finiteness condition πœ‡(𝐸) < +∞, is dropped.

 

 

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