Recent content by sbashrawi

Homework Statement Hi every body I am triyng to find a polynolial approximation to the function: f(x)= (x+2)ln(x+2) using the chebyshev polynomials, the idea is to use MATLAB to find the coeefficients of the approximation poly. using the comand double(int(...)) but this command...

Homework Statement show that if f is increasing on [a, b], then f is absolutely continuous if and only if for each \epsilon > 0 there is a \delta > 0 such that for each measurable subset E of [a, b], m*(f(E)) < \epsilon if m(E) < \delta. Homework Equations The Attempt at a Solution

I know this but the problem is how to prove it. I proved it in the follwoing way: let f be simple function on [ a+h, b+h] f(x) = sum( c_i X(E_i+h)) int f(t) over [a+h, b+h] = sum (c_i * m(E_i + h) = sum (c_i * m(E_i)) = sum (c_i * m( E_i -h)) = int f(t+h) over [a,b]. Am I right?

In fact it is lebesgue integrable function and this is part of a problem. The problem asked to show this property for simple integrable function over [ a +h, b+h], then proceed to prove the general case.

Homework Statement What are the upper derivative and lower derivative of the characteristic function of rationals? Homework Equations The Attempt at a Solution I think they are : upper derivative = 0 lower derivative = negative infinity

I am sorry, You are right. Thank you

I don't think this works , since you used two dumy ( u_ 1, u_2) variables to find the limits and we are supposed to use just one

No, I don't think: g( t) : [a+h, b+h] g(t+h) : [a, b]. substitution will give : let x = t+h , then t = x-h which is defferent from the limits we have

Homework Statement I need to proof the change of variable formula for integartion integration of [g(t)]dt on [a+h, b+h] =integration of g(t+h)dt on [ a, b] Homework Equations The Attempt at a Solution

Homework Statement Let f be continuous on R. Is there an open interval on which f is monotone? Homework Equations The Attempt at a Solution I think there is such interval for non constant function but I am really not sure.

A family F of measurable functions is tight on E if there is a measurable subset E1 of finite measure such that integration of |fn| on ( E-E1) is less than epsilon for each fn in F

Homework Statement If for each \epsilon>0 , there is ameasurable subset E1 of E that has finite measure and a \delta>0 such that for each measurable subset A of E and index n if m(A\capE1) < \delta , then \int | fn| <\epsilon ( integration over A) Show that {fn} is tight...

I mean lebesgue measure

They are measurable since they are inverses of borel sets ( intervals)

Homework Statement If E is a measurable set of measure zero, and f is bounded function on E. Is f measurable? I tried to prove this by saying that E = { x in E | m< f(x) <M} = {x in E | f(x) > m }intersecting { x in E | f(x) < M } and these are measurable so f is measurable. Am I right...