Integration Theory-I

(i) Which of the following is incorrect:

(a) IR is measurable set

(b) E is measureable set if μ (E) = 0

(c) If A ⊂ B and μ (A) < ∞, then μ (B−A) = μ (B)−μ (A)

(d) None of these

(ii) Which of the following is correct:

(a) |V (E)| = V⁺ (E) - V^- (E) ∀ E ∈ M

(b) |V (E)| ≤ |V| (E) ∀ E ∈ M

(c) Both (a) and (b)

(d) None of these

(iii) If μ: S → [0, ∞] is a premeasure, where S be a collection of subsets of a set X. Then for φ ∈ S

(a) μ (φ) = 0

(b) μ (φ) ≥ 0

(c) μ (φ) ≤ 0

(d) None of these

(iv) If f and g are measureable real valued functions. Then, which of the following is correct :

(a) f + g is measurable

(b) f.g is measurable

(c) max {f, g} is measurable

(d) None of the above

(v) If f is a nonnegative measurable function on x, then

(a) ∫ f dμ = 0 <=> f = 0 a.e on x

(b) ∫ f = 0 a.e on x => ∫ f dμ = 0

(c) ∫ f dμ = 0 <=> f = 0 a.e on x

(d) All of the above

Define complete measure space with example.

Define finite and σ-finite measure with example.

Define the following :-

(i) Hahn decomposition

(ii) Jordan decomposition

Define the following :-

(i) Signed measure

(ii) Mutually singular measure

Show that any measure that is induced by an outer measure is complete.

Define premeasure with example.

Define measurable function with example.

Let (x, M, μ) be a measurable space and {f_n} be a sequence of measurable functions on x. Then prove that lim sup {f_n} and lim inf {f_n} are measurable.

Let (X, M, μ) be a measure space and f a nonnegative measurable function on X for which ∫ f dμ < ∞. Then prove that f is finite a.e. on X.

Define the following :-

(i) Simple function

(ii) Integrable function

Let (X, M) be a measurable space. If μ and ν are measures on M and μ ≥ ν, then prove that there is a measure λ on M for which μ = ν + λ.

State and prove Borel Contelli Lemma.

Show that if v₁, and v₂ are any two finite signed measures, then so is αv₁ + βv₂, where α and β are real numbers. Show that measurable function on X. If α and β are positive real numbers then prove that

∫ [α.f + β.g] dμ = α.∫ f dμ + β.∫ g dμ

Show that a set function is a premeasure if it has an extension that is a mensure.

State and prove simple Approximation Lemma.

Prove Egoroff's theorem. Is Egoroff's theorem true in the absence of the assumption that the limit function is finite a.e.?

State and prove Monotone Convergence theorem.

Let (X, M, μ) be a measure space and f and g are nonnegative measureable functions on X. If α and β are positive real numbers then prove that

∫ [α.f + β.g] dμ = α.∫ f dμ + β.∫ g dμ