Let f be a nonnegative measurable function. Show that (integral f = 0) implies f = 0 a.e.
See attached document for notations.
Please help: This problem is from Royden's Chap 4 text on Lebesgue Integral.

Prove theorem 7.3 in notes attached.
Section 7: The LebesgueIntegral
Definition 7.1 Let L be the set of real-valued functions f such that for some g and h in f=g-h almost everywhere. The set L is called the set of Lebesgue integrable function on and the Lebesgueintegral of f is defined as follows: .
Theorem 7

Let f be a nonnegative integrable function. Show that the function F defined by F(x)= Integral[from -inf to x of f] is continuous by using the Monotone Convergence Theorem.
From Royden's RealAnalysis Text, chapter 4.
See the attached file.

Let f be defined on the interval [0,1] by setting f(x)=0 if x is irrational, and if x=(m/n) rational where gcd(m,n)=1 set f(x)=n. Show that f is unbounded on every open interval in [0,1] and compute the Lebesgueintegral of f over the interval [0,1].

Let f be the following function with domain C = [0, 1] X [0, 1] (in two-dimensional Cartesian space):
f(x, y) = 0 on the line segments x = 0, y = 0, and x = y
f(x, y) = -1/(x^2) if 0 < y < x <= 1
f(x, y) = 1/(y^2) if 0 < x < y <= 1
Compute each iterated Riemann integral of f on C (by integrating first over x and then

Using the definition of Lebesgue sum, show Lebesgueintegral.
21.1 Definition: If f is bounded measurable function on a bounded measurable set , if is a partition of and if for then we call a Lebesgue sum of f relative to P.
26.5 Show directly from Definition 21.1 that if f is bounded and m(A)=0, then .
26.7 S

For what values of a in R (real numbers) is the function (1+x^2)^a in L^2
NOTE: let L^2 be like in Lebesgue integration where the set of all measurable functions that are square-integrable forms a Hilbert space, the so-called L2 space.
keywords: integration, integrates, integrals, integrating, double, triple, multiple, r

Prove or disprove the following:
If f is in L^1[0,1], then limit the integral over [0,1] of x^n*f = 0 as n goes to infinity.
I saw a similar example asking to prove that the integral from 0 to 1 of x^2n f(x) dx = 0, and they used algebra of functions generated by {1,x^2}, but we haven't talked about that, so please when you

Show that the sum and product of two simple functions are simple.
Show that
[Definition of simple: A real-valued function is called simple if it is measurable and assumes only a finite number of values. If is simple and ahs the values then , where .]
This problem is from Royden's RealAnalysis text for gradu

Recall that the support of a function f : Rn â?'R is the closure of the
set {x É? Rn : f(x) = 0}. Prove that if f : Rnâ?' Rn has support in a
set of Lebesgue measure =0, then â?«Rn f(x)dx = 0.
The original problem is written in PDF file sent as attachment below. It is number 3 on the list.