Integrable Functions

Question 1

f:X -> R is Lebesgue integrable

(or just integrable)

if…

Answer 1

there exists upper functions u, v such that

f = u - v a.e.

(then ∫f = ∫u - ∫v)

Question 2

f, g integrable means..

Answer 2

f + g, |f|, cf + cg, and max{f, g}

are all integrable!

Question 3

what’s the canoncial way to write

f an integrable function?

Answer 3

f = f⁺ - f^-

b/c f⁺, f^- are upper functions

Question 4

If f =g a.e. and f is integrable then…

Answer 4

g is integrable too!

Question 5

integrable implies

integrable does not imply

Answer 5

implies measurable

does not imply upper!

Question 6

For f integrable, when is f upper?

Answer 6

if f ≥ 0 a.e.

then it’s upper

Question 7

What’s the relationship between f integrable

and measurable sets?

Answer 7

for any epsilon,

A = {x | |f(x)| ≥ epsilon}

is measurable and has finite measure

Question 8

|f| integrable/measurable does not imply

Answer 8

f is not necessarily

integrable or measurable

Question 9

If f is measurable and sandwiched between

two integrable functions….

Answer 9

f is integrable!

Question 10

{x | |f(x)| > e} has finite measure when?

Answer 10

when f is integrable

Question 11

Levi’s Theorem says

Answer 11

there’s an integrable function waiting at the top

of a sequence of integrable functions with

if lim ∫ fn < ∞

Question 12

What does Fatou’s Lemma say about

fn integrable ≥ 0 a.e. ?

Answer 12

lim inf ∫ fn < ∞

then

∫ lim inf fn ≤ lim inf ∫ fn

“take lim inf in and out”

Question 13

What does Lebesgue Dominated Convergence say?

Answer 13

If fn –> f a.e.

and

|fn| < g a.e. for g integrable

then

∫ f = lim ∫ fn

(and both are integrable)