Let
and
be two finite measures on
.
Prove that
if and only if the condition
implies
, for each
.
Thank you for your explanations.

Let and be two finite measures on . Prove that if and only if the condition...
For each
. Find the intersection of and prove.
Please show and explain steps.
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Define
a prime number,
a finite group,
as a Sylow
-subgroup of
.
Assume there exists
a proper subgroup of
where
, i.e. the normaliser of
in
is a subgroup of
.
Prove that
isn't normal in
.
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Prove, or give a counter example to disprove the following
statements.
a)
b)
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Suppose that
is a bounded function with following Lower and Upper
Integrals:
and
a) Prove that for every
, there exists a partition
of
such that the difference between the upper and lower sums
satisfies
.
b) Furthermore, does there have to be a subdivision such that
. Either prove it or find a counterexample and show to the
contrary.
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Prove the following
Let
with
Then:
i)
if and only if
where the double inequality
means
and
ii) If
,
if and only if
.
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Solve the harmonic oscillator motion for initial conditions x(0)
= 0, V(0) = V0 in the case of (a) underdamped
(b) overdamped
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Let X be a banach space such that X= C([a,b]) where - ab+ with the sup
norm. Let x and f X. Show
that the non linear integral equation
u(x) = (sin
u(y) dy + f(x) ) has a solution u X. (the integral is
from a to b).
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Let
be a sequence of random variables, and let Y be a random
variable on the same sample space. Let An(ϵ) be the
event that |Yn − Y | > ϵ. It can be shown that a
sufficient condition for Yn to converge to Y w.p.1 as
n → ∞ is that for every ϵ > 0,
(a) Let
be independent uniformly distributed random variables on [0, 1],
and let Yn = min(X1, . . . , Xn).
In class,...
Prove the ratio test . What does this tell you if
exists?
(Ratio test) If
for all sufficiently large n and some
r < 1, then
converges absolutely; while if
for
all sufficiently large n, then
diverges.
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A Pareto distribution is often used in economics to explain a
distribution of wealth. Let a random variable X have a Pareto
distribution with parameter θ so that its probability distribution
function is
for
and 0 otherwise. The parameters and
are
known and fixed; is a constant to
be determined.
a) Assuming that
find the expected value and variance of ?
b) Show that for 3 ≥ θ > 2 the Pareto distribution has a
finite mean but infinite variance,...