Question

Let f be a function with measurable domain D. Then f is measurable if and only if the function g(x)={f(x) if x\in D ,0 if x \notin D } is measurable.

Answer #1

HHere is the solution

A function f is said to be Borel measurable provided its domain
E is a Borel set and for each c, the set {x in E l f(x) > c} is
a Borel set. Prove that if f and g are Borel measurable functions
that are defined on E and are finite almost everywhere on E, then
for any real numbers a and b, af+bg is measurable on E and fg is
measurable on E.

Let (X, A) be a measurable space and f : X → R a function.
(a) Show that the functions f 2 and |f| are measurable whenever
f is measurable.
(b) Prove or give a counterexample to the converse statement in
each case.

Let f and g be measurable unsigned functions on R^d . Assume
f(x) ≤ g(x) for almost every x. Prove that the integral of f dx ≤
Integral of g dx.

A function f on a measurable subset E of
Rd is measurable if for all a
in R, the set
f-1([-∞,a)) = {x in
E: f(x) < a} is
measurable
Prove or disprove the following functions are measurable:
(a) f(x) = 8
(b) f(x) = x + 2
(c) f(x) = 3x
(d) f(x) = x2

Let (X , X) be a measurable space. Show that f : X → R is
measurable if
and only if {x ∈ X : f(x) > r} is measurable for every r ∈
Q.

A function f on a measurable subset E of
Rd is measurable if for all a
in R, the set
f-1([-∞,a)) = {x in
E: f(x) < a} is
measurable
Prove that if f is continuous on
Rd then f is measurable

Problem 2. Let F : R
→ R be any function (not necessarily measurable!).
Prove that the set of points x ∈ R such
that
F(y) ≤ F(x) ≤
F(z)
for all y ≤ x and z ≥ x is
Borel set.

Let the function f and g be defined as f(x) = x/ x − 1 and g(x)
= 2 /x +1 . Compute the sum (f + g)(x) and the quotient (f/g)(x) in
simplest form and describe their domains.
(f + g )(x) =
Domain of (f+g)(x):
(f/g)(x) =
Domain of (f/g)(x):

Let Let A = {a, e, g} and B = {c, d, e, f, g}. Let f : A → B and
g : B → A be defined as follows: f = {(a, c), (e, e), (g, d)} g =
{(c, a), (d, e), (e, e), (f, a), (g, g)}
(a) Consider the composed function g ◦ f.
(i) What is the domain of g ◦ f? What is its codomain?
(ii) Find the function g ◦ f. (Find...

4. Let f be a function with domain R. Is each of the following
claims true or false? If it is false, show it with a
counterexample. If it is true, prove it directly from the FORMAL
DEFINITION of a limit.
(a) IF limx→∞ f(x) = ∞ THEN limx→∞ sin (f(x)) does
not exist.
(b) IF f(−1) = 0 and f(1) = 2 THEN limx→∞ f(sin(x)) does not
exist.

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