Question

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

Answer #1

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.

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].

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

Show E[f(X)g(X)]≥E[f(X)]E[g(X)] for f,g bounded,
nondecreasing.

suppose f is an integral element function on [0,1], p and q are
partitions of [0,1].
if Up-Lp <= Uq-La, then q belongs to p.
is this statement true or false? either provide a proof or
counterexample.

Let f : R → R be a continuous function which is periodic. Show
that f is bounded and has at least one fixed point.

Let f : R → R be a bounded differentiable function. Prove that
for all ε > 0 there exists c ∈ R such that |f′(c)| < ε.

Let E, F, G be three events. Find P{E ∪ F ∪ G} function of P{E},
P{F}, P{G}, P{E ∩ F}, P{E ∩ G}, P{F ∩ G}, and P{E ∩ F ∩ G}.

Let f : E → R be a differentiable function where E = [a,b] or E
= (−∞,∞), show that if f′(x) not = 0 for all x ∈ E then f is
one-to-one, i.e., there does not exist distinct points x1,x2 ∈ E
such that f(x1) = f(x2). Deduce that f(x) = 0 for at most one
x.

suppose f is an integral element function on [0,1], p and q are
partitions of [0,1].
for any partitions A and B, let P = A U B. Then Up-Lp <=
Ua-La.
is this statement true or false? either provide a proof or
counterexample.
cross "element"

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