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/Part 4: Measure Theory and Integration/Chapter 27: The Lebesgue Integral

27.3 Integration of Complex-Valued Functions

Definition 27.3.1 (Integrable).label Let $(X, \cm, \mu)$ be a measure space and $f: X \to \complex$ be a $(\cm, \cb_{\complex})$-measurable function, then $f$ is integrable if

\[\int \abs{f}d\mu < \infty\]

The set

\[\mathcal{L}^{1}(X, \cm, \mu; \complex) = \mathcal{L}^{1}(X, \cm, \mu) = \mathcal{L}^{1}(X; \complex) = \mathcal{L}^{1}(X) = \mathcal{L}^{1}(\mu; \complex) = \mathcal{L}^{1}(\mu)\]

is the vector space of $\mu$-integrable functions on $X$.

Proof. Let $f, g \in \mathcal{L}^{1}(X)$ and $\lambda \in \complex$, then

\[\int \abs{\lambda f + g}d\mu \le \int \abs{\lambda}\abs{f}+ \abs{g}d\mu = \lambda \int \abs{f}d\mu + \int \abs{g}d\mu\]

by Proposition 27.2.7.$\square$

Definition 27.3.2 (Positive and Negative Parts).label Let $X$ be a set and $f: X \to \real$ be a function, then

\[f^{+} = f \vee 0 \quad f^{-} = -(f \wedge 0)\]

are the positive and negative parts of $f$, and $f = f^{+} - f^{-}$.

Definition 27.3.3 (Integral).label Let $(X, \cm, \mu)$ be a measure space and $f \in \mathcal{L}^{1}(X)$. If $f$ is $\real$-valued, then

\[\int f d\mu = \int f^{+} d\mu - \int f^{-} d\mu\]

is the integral of $f$. If $f$ is $\complex$-valued, then

\[\int f d\mu = \int \text{Re}(f)d\mu + i\int \text{Im}(f)d\mu\]

is the integral of $f$.

Proposition 27.3.4 ([Proposition 2.21, 2.22, Fol99]).label Let $(X, \cm, \mu)$ be a measure space, then the integral is a linear functional on $\mathcal{L}^{1}(X)$ such that for any $f \in \mathcal{L}^{1}(X)$, $\abs{\int f d\mu}\le \int \abs{f}d\mu$.

Proof. By Lemma 17.3.7, the mapping

\[I: \mathcal{L}^{1}(X; \real) \to \real \quad f \mapsto \int f^{+} d\mu - \int f^{-} d\mu\]

is a $\real$-linear functional on $\mathcal{L}^{1}(X; \real)$.

Let $f, g \in L^{1}(X; \complex)$ and $\lambda = \alpha + \beta i \in \complex$ with $\alpha, \beta \in \real$, then

\begin{align*}\int (\alpha f)d\mu&= \int \text{Re}(\lambda f)d\mu + i\int \text{Im}(\lambda f)d\mu \\&= \int \alpha \text{Re}(f) - \beta \text{Im}(f)d\mu + i\int \beta\text{Re}(f) + \alpha \text{Im}(f)d\mu \\&= \alpha \braks{\int \text{Re}(f)d\mu + i\int \text{Im}(f)}d\mu + i\beta \braks{\int \text{Re}(f)d\mu + i\int \text{Im}(f)}d\mu \\&= \lambda \int f d\mu\end{align*}

and

\begin{align*}\int f + g d\mu&= \int \text{Re}(f + g)d\mu + i\int \text{Im}(f + g)d\mu \\&= \int \text{Re}(f)d\mu + i\int\text{Im}(f)d\mu + \int \text{Re}(g)d\mu + i\int \text{Im}(g)d\mu \\&= \int f d\mu + \int g d\mu\end{align*}

so the integral is a linear functional on $\mathcal{L}^{1}(X)$.

Finally, if $f$ is $\real$-valued, then

\[\int f = \int f^{+} d\mu - \int f^{-}d\mu \le \int f^{+} + \int f^{-}d\mu = \int \abs{f}d\mu\]

and if $f$ is $\complex$-valued and $\alpha = \ol{\sgn(\int f d\mu)}$, then

\begin{align*}\abs{\int f d\mu}&= \alpha \int f d\mu = \int \alpha f \\&= \text{Re}\paren{\int \alpha f d\mu}= \int \text{Re}(\alpha f)d\mu \\&\le \int \abs{\text{Re}(\alpha f)}d\mu \le \int \abs{\alpha f}d\mu = \int \abs{f}d\mu\end{align*}

so $\abs{\int f d\mu}\le \int \abs{f}d\mu$.$\square$

Remark 27.3.1.label The construction of the Lebesgue integral using positive/negative and real/imaginary parts, and the deduction of properties from this definition both appear cumbersome. This comes from the lack of usage of its linearity on integrable simple functions. A more functional analytic approach will use this linearity and the density of simple functions to construct the integral. This is the route through which the Bochner integral will be presented.

Theorem 27.3.5 (Dominated Convergence Theorem).label Let $(X, \cm, \mu)$ be a measure space, $\seq{f_n}\subset \mathcal{L}^{1}(X)$, and $f:X \to \complex$ such that

  1. (1)

    $f_{n} \to f$ pointwise.

  2. (2)

    There exists $g \in \mathcal{L}^{1}(X)$ such that $\abs{f_n}\le \abs{g}$ for all $n \in \natp$.

then $\int fd\mu = \limv{n}\int f_{n} d\mu$.

Proof [Theorem 2.24, Fol99]. By (1) and (2), $f$ is measurable with $\int \abs{f}d\mu \le \int \abs{g}d\mu < \infty$, so $f \in \mathcal{L}^{1}(X)$. Now, since $g + f, g - f \ge 0$, by Fatou’s lemma and Proposition 27.3.4,

\begin{align*}\int g d\mu + \int f d\mu&\le \liminf_{n \to \infty}\int g + f_{n} d\mu = \int g d\mu + \liminf_{n \to \infty}f_{n} d\mu \\ \int g d\mu - \int f d\mu&\le \liminf_{n \to \infty}\int g - \int f_{n} d\mu = \int g d\mu - \limsup_{n \to \infty}\int f_{n} d\mu\end{align*}

so

\[\limsup_{n \to \infty}\int f_{n} d\mu \le \int f d\mu \le \liminf_{n \to \infty}\int f_{n} d\mu\]

and $\int f d\mu = \limv{n}\int f_{n} d\mu$.$\square$

Remark 27.3.2 (Dominated Convergence Theorem for Nets?).label In analysis, one frequently encounters places where only sequential continuity is provided or required. It is my opinion that a good portion of this comes from the lack of an extension of the dominated convergence theorem to nets. This limitation arises from the monotone convergence theorem, where continuity from below is used.

For an example, consider the Lebesgue measure on $[0, 1]$. Let $A$ be the net of all finite subsets of $[0, 1]$, directed by inclusion, then $\lim_{\alpha \in A}\one_{\alpha} = 1$ pointwise. However, $\int \one_{\alpha} = 0$ for all $\alpha \in A$.

At least, that is how I thought back in 2025. While arbitrary pointwise convergence has no reason to cooperate with the structure of a measure space, additionally supplying convergence in measure bridges the above gap. The corresponding Monotone Convergence Theorem is given at Theorem 26.8.4, and the Dominated Convergence Theorem is given at Corollary 16.4.4 following the Vitali Convergence Theorem.

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