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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.436J/15.085J
Fall 2008
Problem Set 7
due 10/29/2008
Readings:
Notes for lectures 1113 (you may skip the proofs in the notes for
lecture 11).
Optional additional readings:
∞
Adams & Guillemin, Sections 2.22.3, skim Section 2.5.±
For a full development of this material, see [W], Sections 5.15.9, 6.06.3, 6.5,±
6.12, 8.08.4.±
Exercise 1.
Show that if
g
: Ω
→
[0
,
∞
]
satisﬁes
g dµ
<
∞
, then
g <
∞
,
a.e. (i.e., the set
{
ω

g
(
ω
) =
∞}
has zero measure).
Exercise 2.
Let
(Ω
,
F
,
P
)
be a probability space. Let
g
: Ω
→
R
be a nonneg²
ative measurable function. Let
λ
be the Lebesgue measure. Let
f
be a nonneg²
∞
ative measurable function on the real line such that
f
dλ
= 1
. For any Borel
set
A
, let
P
1
(
A
) =
A
f
dλ
. Prove that
P
1
is a probability measure.
Exercise 3. (Impulses and Impulse Trains)
Consider the real line, endowed with the Borel
σ
ﬁeld. For any
c
∈
R
, we deﬁne
the Dirac measure (“unit impulse”) at
c
, denoted by
δ
c
, to be the probability
measure that satisﬁes
δ
c
(
c
) = 1
. If we “place a Dirac measure” at each integer,
n
=1
n
=1
we are led to the measure
µ
=
, that is,
µ
(
A
) =
(
A
)
, for every±
δ
n
δ
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 Fall '08
 DavidGarnarnik
 measure, Lebesgue integration, Borel, nonnegative random variables, lim aij

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