Integration of Complex Functions

As before, $\mu$ will in this section be a positive measure on an arbitrary measurable space $X$ .

1.30 Definition We define $L^1(\mu)$ to be the collection of all complex measurable functions $f$ on $X$ for which

Note that the measurability of $f$ implies that of $|f|$ , as we saw in Proposition 1.9(b); hence the above integral is defined.

The members of $L^1(\mu)$ are called Lebesgue integrable functions (with respect to $\mu$ ) or summable functions. The significance of the exponent 1 will become clear in Chap. 3.

1.31 Definition If $f = u + iv$ , where $u$ and $v$ are real measurable functions on $X$ , and if $f \in L^1(\mu)$ , we define

for every measurable set $E$ .

Here $u^+$ and $u^-$ are the positive and negative parts of $u$ , as defined in Sec. 1.15; $v^+$ and $v^-$ are similarly obtained from $v$ . These four functions are measurable, real, and nonnegative; hence the four integrals on the right of (1) exist, by Definition 1.23. Furthermore, we have $u^+ \le |u| < |f|$ , etc., so that each of these four integrals is finite. Thus (1) defines the integral on the left as a complex number.

Occasionally it is desirable to define the integral of a measurable function $f$ with range in $[-\infty, \infty]$ to be

provided that at least one of the integrals on the right of (2) is finite. The left side of (2) is then a number in $[-\infty, \infty]$ .

1.32 Theorem Suppose $f$ and $g \in L^1(\mu)$ and $\alpha$ and $\beta$ are complex numbers. Then $\alpha f + \beta g \in L^1(\mu)$ , and

PROOF The measurability of $\alpha f + \beta g$ follows from Proposition 1.9(c). By Sec. 1.24 and Theorem 1.27,

Thus $\alpha f + \beta g \in L^1(\mu)$ .

To prove (1), it is clearly sufficient to prove

and

and the general case of (2) will follow if we prove (2) for real $f$ and $g$ in $L^1(\mu)$ .

Assuming this, and setting $h = f + g$ , we have

or

By Theorem 1.27,

and since each of these integrals is finite, we may transpose and obtain (2).

That (3) holds if $\alpha \ge 0$ follows from Proposition 1.24(c). It is easy to verify that (3) holds if $\alpha = -1$ , using relations like $(-u)^+ = u^-$ . The case $\alpha = i$ is also easy: If $f = u + iv$ , then

Combining these cases with (2), we obtain (3) for any complex $\alpha$ . ////

1.33 Theorem If $f \in L^1(\mu)$ , then

PROOF Put $z = \int_X f \, d\mu$ . Since $z$ is a complex number, there is a complex number $\alpha$ , with $|\alpha| = 1$ , such that $\alpha z = |z|$ . Let $u$ be the real part of $\alpha f$ . Then $u \le |\alpha f| = |f|$ . Hence

The third of the above equalities holds since the preceding ones show that $\int \alpha f \, d\mu$ is real. ////

We conclude this section with another important convergence theorem.

1.34 Lebesgue’s Dominated Convergence Theorem Suppose $\{f_n\}$ is a sequence of complex measurable functions on $X$ such that

exists for every $x \in X$ . If there is a function $g \in L^1(\mu)$ such that

then $f \in L^1(\mu)$ ,

and

PROOF Since $|f| \le g$ and $f$ is measurable, $f \in L^1(\mu)$ . Since $|f_n - f| \le 2g$ , Fatou’s lemma applies to the functions $2g - |f_n - f|$ and yields

Since $\int 2g \, d\mu$ is finite, we may subtract it and obtain

If a sequence of nonnegative real numbers fails to converge to 0, then its upper limit is positive. Thus (5) implies (3). By Theorem 1.33, applied to $f_n - f$ , (3) implies (4). ////