Question

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.

Answer #1

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 (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.

Let D ⊆ R, a ∈ D, let f, g : D −→ R be continuous functions. If
limx→a f(x) = f(a) and limx→a g(x) = g(a) with f(a) < g(a), then
there exists δ > 0 such that x ∈ D, 0 < |x − a| < δ =⇒
f(x) < g(x).

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.

Let f and g be continuous functions on the reals and let S={x in
R | f(x)>=g(x)} . Show that S is a closed set.

[Lebesgue measurability] Can anyone prove this?
Let f : Rd → [0, ∞] be an unsigned function. Prove that
the function f is measurable.
It
is just an unsigned function. The question does not talk about if f
is continuous or not. Thanks.

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 f and g be functions between A and B. Prove that f = g iff
the domain of f = the domain of g and for every x in the domain of
f, f(x) = g(x).
Thank you!

Let f and g be continuous functions from C to C and let D be a
dense
subset of C, i.e., the closure of D equals to C. Prove that if
f(z) = g(z) for
all x element of D, then f = g on C.

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

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