Get premium membership and access revision papers, questions with answers as well as video lessons.

Sma2110:Real Analysis Ii Question Paper

Sma2110:Real Analysis Ii

Course:Bachelor Of Computer Science

Institution: Meru University Of Science And Technology question papers

Exam Year:2012

QUESTION ONE (30 MARKS)
(a) Define a algebra and give an example. (4 Marks) (b) Show that a -algebra is closed under countable intersection. (5 Marks) (c) Define a measure space. (2 Marks) (d) (i) Define a measurable set. (1 Mark) (ii) Show that if is zero, then E is L measurable. (4 Marks) (e) Show that the characteristic function on a measurable set is measurable. (4 Marks) (f) (i) Define a probability measure. (2 Marks) (ii) Why is it called a normalized measure? (2 Marks) (g) Let be a simple function belonging to +,. Let : be defined by = for Show that is a measure. (3 Marks) (h) State the monotone convergence theorem. (2 Marks) (i) When is a sequence of measureable functions said to converge to a function (2 Marks)
2
QUESTION TWO (20 MARKS)
(a) What do you understand by the Lebesgue outer measure of a subset (3 Marks) (b) Show that = 0 (3 Marks)
(c) Let = 0 1and1,2, 0 1 be a finite collection of points. Define a set function : 0 8 be such that , where , is the number of points in the collection that lie in E. Is an outer measure? (6 Marks)
(d) one that contains no sevens in any decimal expansion. Compute the Lebesgue measure of the totally unlucky numbers in0 1. (6 Marks)
QUESTION THREE (20 MARKS)
(a) Let f be a differentiable function in . Show that f is measurable. (5 Marks) (b) Let f be a function on for which is measurable. Can we conclude that f is measurable? If not give an example. (5 Marks) (c) Let be a sequence of x measurable function . Show that the set of all points over which converges pointwise is measurable. (6 Marks) (d) State Fatuous Lemma. (4 Marks)
QUESTION FOUR (20 MARKS)
(a) Write the Canonical representation of a simple function. (3 Marks)
(b) When do we say that a property holds in . (3 Marks)
(c) Let and 1,2 + such that 1 < and 2 < suppose = 2 1. Show that = 2 . (5 Marks)
(d) Let be a sequence of real valued, functions on a measure space ,,. Suppose that the sequence conveys pointwise to a function and is dominated by some intangible function g in the sense that and all points in X. Prove that f is integrable and ? ? ? ?? s s n fd df n ?? lim (9 Marks)

Sma2110:Real Analysis Ii Question Paper

More Question Papers

Popular Exams