9.7. THE LEBESGUE INTEGRAL, L1 207

9.7 The Lebesgue Integral, L1

The functions considered here have values in C, which is a vector space. A function f withvalues in C is of the form f = Re f + i Im f where Re f and Im f are real valued functions.In fact Re f = f+ f

2 , Im f = f− f2i .

Definition 9.7.1 Let (Ω,S ,µ) be a measure space and suppose f : Ω→ C. Thenf is said to be measurable if both Re f and Im f are measurable real valued functions.

Of course there is another definition of measurability which says that inverse images ofopen sets are measurable. This is equivalent to this new definition.

Lemma 9.7.2 Let f : Ω→ C. Then f is measurable if and only if Re f , Im f are bothreal valued measurable functions. Also if f ,g are complex measurable functions and a,bare complex scalars, then a f +bg is also measurable.

Proof: ⇒Suppose first that f is measurable. Recall that C is considered as R2 with(x,y) being identified with x+ iy. Thus the open sets of C can be obtained with either of

the two equivlanent norms |z| ≡√(Rez)2 +(Imz)2 or ∥z∥

∞= max(Rez, Imz). Therefore,

if f is measurable Re f−1 (a,b)∩ Im f−1 (c,d) = f−1 ((a,b)+ i(c,d)) ∈F . In particular,you could let (c,d) = R and conclude that Re f is measurable because in this case, theabove reduces to the statement that Re f−1 (a,b) ∈F . Similarly Im f is measurable.⇐ Next, if each of Re f and Im f are measurable, then

f−1 ((a,b)+ i(c,d)) = Re f−1 (a,b)∩ Im f−1 (c,d) ∈F

and so, since every open set is the countable union of sets of the form (a,b)+ i(c,d) , itfollows that f is measurable.

Now consider the last claim. Let h : C×C→ C be given by h(z,w) ≡ az+bw. Thenh is continuous. If f ,g are complex valued measurable functions, consider the complexvalued function, h◦ ( f ,g) : Ω→ C. Then

(h◦ ( f ,g))−1 (open) = ( f ,g)−1 (h−1 (open))= ( f ,g)−1 (open)

Now letting U,V be open in C, ( f ,g)−1 (U×V ) = f−1 (U)∩ g−1 (V ) ∈ F . Since ev-ery open set in C×C is the countable union of sets of the form U ×V, it follows that( f ,g)−1 (open) is in F . Thus a f +bg is also complex measurable. ■

As is always the case for complex numbers, |z|2 = (Rez)2 +(Imz)2. Also, for g a realvalued function, one can consider its positive and negative parts defined respectively as

g+ (x)≡ g(x)+ |g(x)|2

, g− (x) =|g(x)|−g(x)

2.

Thus |g| = g+ + g− and g = g+ − g− and both g+ and g− are measurable nonnegativefunctions if g is measurable.

Then the following is the definition of what it means for a complex valued function fto be in L1 (Ω).