Lebesgue integrals

Compute the quantity limit of ( integral from 0 to 1 e^(-x^2/n) dx) ( the integral here is with respect to Lebesgue measure). Make sure that you verify your manipulations by referring to known theorems.

Lebesgue Integrals

Let a,b be real numbers such that 0 < a < b < infinity. Does the limit lim of ( integral from a to b of n*sin (x^2/n) dx , n is positive integer. exist? ( prove or disprove). Find the limit if it exists. Prove all assertions and justify every step. The integral here is with respect of Lebesgue measure.

Lebesgue integrals

Let {f_n} be a sequence of nonnegative Lebesgue measurable functions on [0,1]. Suppose that: (i) f_n -> f in [0,1] and (ii) integral over [0,1] of f_n =< K for all n and some constant K. Then f is in L^1[0,1] and || f||_1 =< K. All integrals are with respect to Lebesgue measure.

Lebesgue measurable sets

Prove or disprove: ( please justify every claim and step) If the boundary of set omega in R^d has an outer measure zero, then omega is Lebesgue measurable.

Lebesgue integrals

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. All integrals here are with respect to Lebesgue measure. Please justify every step and claim. e here is the exponential function.