Real Analysis : Jacobians and Trigonometric Functions

If , find Please see the attached file for the fully formatted problems.

Real Analysis : Elementary Sets and Closure

1). Let M be an elementary set. Prove that | closure(M)M | = 0. ( closure of M can also be written as M bar, and it is the union of M and limit points of M). 2). If M and N are elementary sets then show that | M union N | + | M intersection N| = |M| + |N| The definition of elementary set : If M is a union of finite membe ...continues

Limits : lim n--> ∞ ∫ 0 --> ∞ sin nt/ (1 + t^2) dt

Find the limit and justify your answer: lim n--> ∞ ∫ 0 --> ∞ sin nt/ (1 + t^2) dt Please see the attached file for the fully formatted problems.

Harmonic Analysis, Convolution and L^1

Let be a positive function in . Define a new function by Prove that . Please see the attached file for the fully formatted problems.

Borel Measurable and Borel Functions

1).Let f(X) : R -> R be the following: f(x) = { 1 if x is in Q (rationals) , 0 if x is not in Q ( irrational)} Prove that f(x) is Borel measurable ( Borel functions).

Borel Measurable and Borel Functions

Let f(x) be { 1/x if x is not 0. and 1 if x = 0} . Prove that f(x) is borel function ( borel measurable).

Continuity and Outer Measure

Let f(x) be a positive continuous function on [0,1/2], f(x) =< 1/2. Let A = { (x,y) : 0 =< x = 1/2, 0=

Proof : Outer Measure

Let m'(A) = inf sum of |M_i| where i is from 1 to infinity, such that A is a subset of M_i. M_i's are disjoint. Is m'(A) = m*(A) ? m*(A) is outer measure.

Outer Measure

Let A = union ( i from 1 to infinity) of M_i, Mi's are disjoint, show that m*(A) = sum (i from 1 to infinity) of |M_i| m*(A) is the outer measure of A, that is, m*(A) = inf sum (i from 1 to infinity) of M_i. PLEASE NOTICE THE = SIGN, A = the union, not a subset of the union.

Real Analysis : Jacobians

If and , find the Jacobian of with respect to . Please see the attached file for the fully formatted problems.