Lebesgue Measures and Integrals : Let {f_n} be a sequence of nonnegative Lebesgue measurable functions on [0,1]. Suppose that...

Related BrainMass Solutions

• Lebesgue Measures, Integrals and Limits : Let f_n(x) = n^1/2 * x * e^(-n*x^3), for n = 1,2,3... (i) Find the maximum value assumed by f_n in the interval [0,1] and (ii) Find Lim (n -> infinity) of...

  57868 Lebesgue Measures, Integrals and Limits Let f_n(x) = n^1/2 * x * e^(-n*x^3), for n = 1,2,3... (i) Find the maximum value assumed by f_n in the interval [0,1]. (ii) Find Lim (n -> infinity) of integral from 0 to 1 of (f_n(x))dx.

• Lebesgue Measures and Integrals : Compute the quantity limit of ( integral from 0 to 1 e^(-x^2/n) dx)

  Solution: Suppose that E is a measurable subset of the closed interval [0,1], and let the integrand is a bounded function in then we have: Let fn(x) = Now, using the lebesgue dominated convergence theorem we have

• Q on Lebesgue integrals.

  55373 Q on Lebesgue Integrals In a previous problem I posted here: 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=that; m*(A) = integral from 0 to 1/2 of f(x)dx.

• Reimann and Lebesgue integrals.

  56181 Reimann and Lebesgue integrals. (a) If f is a nonnegative continuous function on [0,1], then show that integral from 0 to 1 f(x) dx = integral over [0,1] f dx ( that is show that the reimann integral and lebesgue integrals are equal).

• Lebesgue Measures, Limits and Integrals : Prove - 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.

  Lebesgue measures and integrals, since all integrals here are with respect to Lebesgue measure.

• Lebesgue Integration

  116031 Lebesgue Integration 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

• Lebesgue integral

  352159 Lebesgue integral 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.

• Real Analysis : Lebesgue Integral Problem

  38810 Real Analysis : Lebesgue Integral Problem Let f be a nonnegative measurable function. Show that (integral f = 0) implies f = 0 a.e. See attached document for notations.

• Real Analysis : Lebesgue Integral and Monotone Convergence Theorem

  38836 Lebesgue Integral and Monotone Convergence Theorem 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.