Measurable Functions

Question 1

a relation holds almost everywhere means

Answer 1

set of points where it fails has measure 0

Question 2

f : x –> R is a measurable function if…

Answer 2

for all open sets O in R,

f^-1(O) is measurable

or

f^-1(a, ∞) is measurable for all a

Question 3

If f is measurable and f = g, a.e. then, …

(f, g: X -> R)

Answer 3

g is measurable

Question 4

What’s a simple example of a measurable function?

Answer 4

constant function

f^-1 is all of X or empty

so it’s measurable!

Question 5

set of inputs where f(x) > g(x)

for f, g measurable

Answer 5

is a measurable set

Question 6

if f, g are measurable then,

+, x, and what else are measurable?

Answer 6

|f(x)|

max{f, g}

and

f⁺ (x) = {f(x) if f(x) >0, else 0}