Integrability Definition and 67 Threads

Homework Statement Let Q=I\times I (I=[0,1]) be a rectangle in R^2. Find a real function f:Q\to R such that the iterated integrals \int_{x\in I} \int_{y\in I} f(x,y) \; and \int_{y\in I} \int_{x\in I} f(x,y) exists, but f is not integrable over Q. Edit: f is bounded Homework...

Hi I am a mathematics junior and I am doing a research project on hamiltonian systems and liouville integrability (don't ask why...). I am using the book by Vilasi, a graduate level book, but I am finding it quite difficult and badly written; for instance he uses functional analysis and...

Homework Statement I've been looking at how integrable functions behave under composition, and I know that if f and g are integrable, f(g(x)) is not necessarily integrable, but it -is- necessarily integrable if f is continuous, regardless of whether g is. So I was wondering, what about if g is...

Hello, question: does the integral \int_0^\infty \frac{\sin(x)}{x^a} converge (in the sense of Lebesgue principal value) for all a \in (0;2)? For a=1/2, it's the Fresnel integral, but other than that, I'm not sure how to approach this.

Homework Statement Let f(x)= { 1 if x=\frac{1}{n} for some n\in the natural numbers, or 0 otherwise} Prove f is integrable on [0,1], and evaluate the integral. Homework Equations This is using Riemann Integrability. I know that the method of providing the solution is supposed to be by...

Homework Statement find the integral of f(x) = x by finding a number A such that L(p,f) <= A <= U(p,f) for all partitions p of [0,1]. where a partition p of an interval [a,b] is of the form {x0,x1, ... , xn} Homework Equations L(p,f) is the lower sum of f with respect to the partition p. In...

f,g integrable on [a,b] how does [tex] \\integral from a to b of (f- lamda*g)^2 = 0 imply (\\integral (fg))^(1/2) =< (\\int f^2)^(1/2)(\\int g^2)^(1/2) ? [\tex] Thank you!

Homework Statement Let f be positive and bounded over [a,b]. If f^2 is integrable over [a,b], then show that f is as well. The Attempt at a Solution I'm just trying to use the fact that the upper and lower sums of f^2 over a partition P are arbitrarily close, and then somehow find an...

f(x) = x , if x is rational = 0 , if x is irrational on the interval [0,1] i just wanted to check if my reasoning is right. take the equipartition of n equal subintervals with choices of t_r's as r/n for each subinterval. calculating the integral as limit of this sum (and...

Q1) Let f(x,y)=3, if x E Q f(x,y)=2y, if y E QC Show that 1 3 ∫ ∫ f(x,y)dydx exists 0 0 but the function f is not (Riemann) integrable over the rectangle [0,1]x[0,3] I proved that the iterated integral exists and equal 9, but I am completely stuck with the second part (i.e. to prove...

It started out as an attempt to solve a HW question (which I also posted in the appropriate forum), but now I'm just curious as to the general case; Assume f>0 is a measurable function from [0,infinity) to itself. Then if xf(x) tends to zero as x tends to zero, there is a positive \epsilon for...

Ok, so I have two questions regarding something I don't understand in my textbook (Adams) 1. 0 if 0<=x<1 or 1<x<=2 f(x) = 1 if x=1 (by "<=" i mean less than or equal) I'm supposed to show that it is Riemann integrable on that interval. They chose P to be: {0, 1-e/3...

hello all Iv been working on a lot of integrability questions and I am having trouble with this problem let f be integrable on [a,b] then show that |f| is integrable and that |\int_{a}^{b}f|\le \int_{a}^{b}|f| now this is what i know \int_{a}^{b^U}f =\int_{a_{L}}^{b}f= \int_{a}^{b}f...

My analysis professor, a few weeks ago, when we were talking about integrability, introduced the concept of Lebesgue measure zero. He put up a theorem stating that the set of discontinuities of a function are of Lebesgue measure zero if and only if the function is integrable. This is, of course...

Hello, If u is a positive measure, I need to show that any finite subset of L^1(u) is uniformly integrable, and if {fn} is a sequence in L^1(u) that converges in the L^1 metric to f in L^1(u), then {fn} is uniformly integrable. I know that a collection of functions {f_alpha}_alpha_in_A...

Let f (x) = 1 if 2<=x<4 2 if x =4 -3, if 4<x<=7 Prove that this function is integrable on [2,7], state its value and prove that it is what you say it is. Obviously integral of f from [2,7] is -7. but its proof and the integrability have me and my friends snagged. Suggestions anyone? SO...

My second course in analysis and i have a problem which i can't understand Let f (x) = 1 if 2<=x<4 2 if x =4 -3, if 4<x<=7 Prove that this function is integrable on [2,7], state its value and prove that it is what you say it is. Obviously integral of f from [2,7]...