3. Give an example of a non-Lebesgue measurable function f: IR→ IR such that || is a measurable function and f((a)) is a measurable set for each a € IR.

For each n E N, let fn → R be a (F,B(R))- measurable function. (i) Then, each of the functions supnen fn, infnen fn, lim supn→∞o fn, and lim infn oo fn is (F, B(R))-measurable. (ii) The set A = {w : limn→∞o fn(w) exists and is finite} lies in F and the function h = (limn→∞ fn) · IA is (F, B(R))-measurable.

Let f A B be a function. Let X be a subset of A. (a) Verify that X ≤ f−¹(ƒ(X)). (b) Give an example of sets X CA and B, and a function f : A → B such that f−¹(f(X)) £ X. (c) Prove that if f is injective then X = f−¹(ƒ(X)).

Q5.3

Consider the following function: f: Z→ N given by f(x) = (x + 4)² + 5; where, as usual, Z and N, denote the set of all integers and the set of all natural numbers, respectively. You are given that the function f is not an injection. In order to show that f is not an injection, state two distinct elements of the domain of f, denoted by ₁, 2, such that f(x₁) = f(x₂). ● State ₁: • State x₂: You are given that the function f is not a surjection. In order to show that f is not a surjection, state an element of the codomain of f, denoted by y, that is not in the image of f. • State y:

1. Define a function f : Z x Z → Z by f(m, n) = 2n – 4m. (a) Prove that ƒ is not one-to-one. (b) Prove that the range of f is equal to the even numbers. In other words, show rng f = {2k|k E Z}, by using the usual methods for proving an equality of sets.

Suppose that E is a measurable set and that f(x)=d for all x∈E. That is, f is a constant function on E. Prove that f is measurable.

Let f be a function from a set X into a set Y. Let E(subalpha) be a subet of Y for each alpha in A. Show f-1(E1UE2) = f-1(E1) U f-1(E2)

Use Stokes's Theorem to evaluate integral C of F · dr.  In this case, C is oriented counterclockwise as viewed from above. F(x, y, z) = z2i + 2xj + y2k S: z = 1 − x2 − y2,  z ≥ 0