Question

Let f be a bounded measurable function on E. Show that there are
sequences of simple

functions on E, {(pn) and {cn}, such that {(pn} is increasing and
{cn} is decreasing and each of

these sequences converges to f uniformly on E.

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 f: R --> R be a differentiable function such that f' is
bounded. Show that f is uniformly continuous.

show that if f is a bounded increasing continuous function on
(a,b), then f is uniformly continuous. Hint: Extend the function to
[a,b].

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 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.

Show that if f is a bounded function on E with[ f]∈ Lp(E), then
[f]∈Lq(E) for all q > p.

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.

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

Let X, Y be metric spaces, with Y complete. Let S ⊂ X and let f
: S → Y be uniformly continuous. (a) Suppose p ∈ S closure and (pn)
is a sequence in S with pn → p. Show that (f(pn)) converges in y to
some point yp.

(IMT 1.1.6).Let E,F⊆R^d be Jordan measurable sets.
1. (Monotonicity) Show that if E⊆F, then m(E)≤m(F).
2. (Finite subadditivity) Show that m(E∪F)≤m(E) +m(F).
3. (Finite additivity) Show that if E and Fare disjoint, then
m(E∪F) =m(E) +m(F).

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