Prove that an outer measure 𝜇∗ on a metric space (𝑋, 𝑑) is a Borel outer measure if and only if 𝜇∗ 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 𝜇∗ on a metric space (𝑋, 𝑑) is a Borel outer measure if and only if 𝜇∗ is a metric outer measure.

 

2. Let 𝜇 be a measure on a 𝜎-algebra 𝒜 of subset of a set 𝑋.

(a) For an arbitrary sequence{𝐸n: 𝑛 ∈ ℕ} in 𝒜, we

have 𝜇(limn→∞𝑖𝑛𝑓 𝐸n) ≤ limn→∞ 𝑖𝑛𝑓𝜇(𝐸n). Give an example where the equality doesn’t hold.

(b)(b) If there exists 𝐴 ∈ 𝒜 with 𝜇(𝐴) < ∞ such that 𝐸n ⊂ 𝐴 for all 𝑛 ∈ ℕ then 𝜇(limn→∞ sup 𝐸n) ≥ limn→∞ 𝑠𝑢𝑝 𝜇(𝐸n). Give an example where the equality doesn’t hold.

(c) If limn→∞ 𝐸n exists and if there exists 𝐴 ∈ 𝒜 with 𝜇(𝐴) < ∞

such that 𝐸n ⊂ 𝐴 for all 𝑛 ∈ ℕ then limn→∞ 𝜇(𝐸n) exists and 𝜇(limn→∞ 𝐸n) = limn→∞ 𝜇(𝐸n).

 

 

 

 

 

 

 

 

Group-B

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

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

2. Let 𝜇 be a measure on a 𝜎-algebra 𝒜 of subset of a set 𝑋.

If {𝐸n: 𝑛 ∈ ℕ} is a decreasing sequence in 𝒜 and if 𝜇(𝐸) < +∞,

then 𝜇(⋂∞ 𝐸n) = limn→∞ 𝜇(𝐸n). Show that equality may fail if

the finiteness condition 𝜇(𝐸) < +∞, is dropped.

 

 

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