Analysis 2
Homework 04
1. For
A
⊂
N
=
{
1
, . . .
} 3
n
let
ν
n
(
A
) =
1
n
μ
#
(
A
∩
[1
, n
]);
let
C
denote the collection of those subsets
A
for which (
ν
n
(
A
))
∞
n
=1
converges
and define
ν
:
C →
[0
,
1] :
A
7→
lim
n
→∞
ν
n
(
A
)
.
Is (
N
,
C
, ν
) a probability space?
2. Let
R
be equipped with the Borel
σ
algebra. Recall that every
continuous
function from
R
to itself is (Borel) measurable. Is every (weakly) monotonic
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 Spring '09
 LARSON
 Calculus, Probability theory, Borel, νn

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