Measures

Question 1

what is a semiring?

Answer 1

nonempty collection of subsets

1) contains empty set
2) A n B in semiring
3) A - B = C₁ u … u C_n

for Cs disjoint members of semiring

Question 2

what is a ring?

Answer 2

nonempty collection of subsets

1) A u B, “+”
2) A - B, “-“

Question 3

what is an algebra?

Answer 3

nonempty collection of subsets

2) A n B
3) A^c

Question 4

what is a σ-algebra?

Answer 4

algebra, closed under

countable U A_i

Question 5

a σ-set of a semiring is…

Answer 5

any countable disjoint union

Question 6

a measure on a semiring S is…

Answer 6

a function u: S -> [0, ∞]

1) u(empty set) = 0
2) u(Union Ai) = u(A1) + u(A2) + …

for Ai countable disjoint elements

and

U(Ai) is in semiring

“countably additive”

Question 7

in semirings , A - countable union of Ai

can be written as

Answer 7

finite disjoint union

of memebers of S

Question 8

σ-set theorems

Answer 8

1) For any {Ai} (not nec disjoint), union Ai is σ-set
2) countable unions and finite n of σ-set

are σ-set

Question 9

A contained in B

implies what about u(A)?

Answer 9

u(A) <= u(B)

Question 10

what are equivalent conditions for a

measure

Answer 10

1) u(Ø) = 0
2) finite union disjoint Ai in A => sum u(Ai) <= u(A)
3) B is in countable union Bi (not nec disjoint)

then u(B) <= sum u(Bi)

Question 11

an outer measure is

Answer 11

a function ü: P(X) -> [0, ∞]

1) ü(Ø) = 0
2) A contained in B => ü(A) <= ü(B)
3) ü(countable union Ai) <= sum ü(Ai)

“countably subadditive”

(Ai not nec disjoint!)

Question 12

A subset E is measurable

(for some outer measure ü)

if…

Answer 12

for all A in X,

ü(A) = ü(A n E) + ü(A n E^c)