Simple and Step Functions

Question 1

A function g: X –> R

is simple if …

Answer 1

g is measurable

and

takes on finitely many values

Question 2

For a_i the distinct, nonzero values of

a simple function g(x),

the standard representation of g(x) is

Answer 2

g(x) = ∑ a_i XAi

where A_i is the set of inputs yielding ai

X is indicator function

Question 3

A simple function is called a

step function if…

Answer 3

u(A_i) < ∞

where A_i are the inputs corresponding to

some nonzero ouput ai

Question 4

Integral of a step function g(x) is…

Answer 4

∑ ai u^*(Ai)

where Ai are inputs

Question 5

What are nice properties of integral of step functions?

Answer 5

∫ f + g = ∫f + ∫ g

for f, g step

and

c∫f = ∫cf

for c in R

Question 6

Do you need the standard rep of a step function

to take it’s integral?

Answer 6

NO!

If f = Σ pi Xi

for any measurable sets pi with u(pi) < ∞,

∫f = Σpi u(pi)

Question 7

If f ≥ g a.e.,

(or f = g a.e.)

then…

Answer 7

∫ f ≥ ∫g

(∫f = ∫g if f = g a.e.)

Question 8

If f_n ↓ 0 a.e., then …

or

if f_n ↑ f a.e., then…

Answer 8

∫ f_n ↓ 0

or

∫ f_n ↑ ∫ f

why? consider f - f_n ↓ 0

*careful with up and down!

Question 9

If fn step ↑ f and gn ↑ f a.e.,

then…

Answer 9

lim ∫fn = lim ∫ gn

(possibly with both = ∞)

Question 10

How can you show a set A is measurable

using step functions?

Answer 10

Find fn step ↑ XA

(this also show lim ∫ fn = u(A))

because X_A measurable

(A = inverse image of 1)

Question 11

How can you approximate a measurable function

with simple functions?

Answer 11

If f is a measurable func

and f(x) ≥ 0 a.e.

then,

fn ↑ f a.e.

with 0 ≤ fn

Question 12

what is a sigma-finite measure space?

Answer 12

If there are measurable sets

E1 ≤ E2 ≤ E3 … where

U Ei = X and u(Ei) < ∞

Question 13

If X is sigma-finite and f is a measurable func ≥ 0 a.e.,

then…

Answer 13

there exists step functions fn ↑ f a.e.