Real and Complex Analysis (Rudin)

ABSTRACT INTEGRATION 13
1.12 Theorem Suppose rol is a u-algebra in X, and Y is a topological space.
Letfmap X into Y.
(a) If n is the collection of all sets E c Y such that f -1(E) E rol, then n is a
u-algebra in Y.
(b) Iffis measurable and E is a Borel set in Y, thenf-1(E) E rol.
(c) If Y = [-00,00] andf-1((IX, 00]) E rolfor every real IX, thenfis measurable.
(d) If f is measurable, if Z is a topological space, if g: Y --+ Z is a Borel
mapping, and if h = 9 0 f, then h: X --+ Z is measurable.
Part (c) is a frequently used criterion for the measurability of real-valued
functions. (See also Exercise 3.) Note that (d) generalizes Theorem 1.7(b).
PROOF (a) follows from the relations
f-1(Y) = X,
f-1(y - A) = X - f- 1(A),
and f-1(A 1 u A2 U ... ) =f-1(A 1) uf-1(A 2) u ....
To prove (b), let n be as in (a); the measurability of f implies that n
contains all open sets in Y, and since n is au-algebra, n contains all Borel
sets in Y.
To prove (c), let n be the collection of all E c [ - 00, 00] such that
f -1(E) E rol. Choose a real number IX, and choose IXn < IX so that IXn --+ IX as
n --+ 00. Since (IXn' 00] E n for each n, since
00 00
[-00, IX) = U [-00, IXn] = U (IXn' 00]