QUIZ 4.
MEASURE AND INTEGRATION
Let (X, C, µ) be a fixed measure space.
(a) Let E and F be C−measurable sets, E ⊂ F . Show that µ(E) ≤
µ(F ). Also show that µ(F \ E) = µ(F ) − µ(E) if µ(E) < +∞.
(b) What does it mean that a function f is equal to 0 µ−almost
everywhere?
(c) What does it mean that a simple function φ is written on standard
form?
(d)How is the integral of a simple measurable (nonnegative) function φ defined, and how can we define the integral of a non-negative
measurable function f ?
1