In: Advanced Math

Prove that f : R → R is Lebesgue measurable if and only if the preimage of every Borel set is a Lebesgue measurable.

Suppose f : R → R is measurable and g : R → R is monotone. Prove
that g ◦ f is measurable

Suppose I is an interval in R. Prove from the definition of
outer Lebesgue measure that m^∗ (I) = I.

Prove that {f(x) ∈ F(R, R) : f(0) = 0} is a subspace of F(R, R).
Explain why {f(x) : f(0) = 1} is not.

Let f : R → R be a function.
(a) Prove that f is continuous on R if and only if, for every
open set U ⊆ R, the preimage f −1 (U) = {x ∈ R : f(x) ∈ U} is
open.
(b) Use part (a) to prove that if f is continuous on R, its zero
set Z(f) = {x ∈ R : f(x) = 0} is closed.

State and prove Lebesgue Dominated Convergence theorem.

prove that a ring R is a field if and only if (R-{0}, .) is an
abelian group

1. Prove or disprove: if f : R → R is injective and g : R → R is
surjective then f ◦ g : R → R is bijective.
2. Suppose n and k are two positive integers. Pick a uniformly
random lattice path from (0, 0) to (n, k). What is the probability
that the first step is ‘up’?

a) Suppose f:R → R is differentiable on R. Prove that if f ' is
bounded on R then f is uniformly continuous on R.
b) Show that g(x) = (sin(x4))/(1 + x2) is
uniformly continuous on R.
c) Show that the derivative g'(x) is not bounded on R.

Prove that {??+?:?,?∈?} is dense in ? if and only
if r is an irrational number.

Prove that {??+?:?,?∈?} is dense in ? if and only
if r is an irrational number.

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