Integrating complex functions. Let \((X, \mathscr{A}, \mu)\) be a measure space. For a complex number \(z=x+i y \in \mathbb{C}\) we write \(x=\operatorname{Re} z\) and \(y=\operatorname{Im} z\) for the real and imaginary parts, respectively; \(\mathscr{O}_{\mathbb{C}}\) denotes the usual (Euclidean) topology given by the norm \(|z|=\sqrt{x^{2}+y^{2}}\). Finally, set \(g: \mathbb{C} ightarrow \mathbb{R}^{2}, z=x+i y \mapsto(x, y)\).

(i) \(\mathscr{C}:=g^{-1}\left(\mathscr{B}\left(\mathbb{R}^{2}ight)ight):=\left\{g^{-1}(B): \mathscr{B}\left(\mathbb{R}^{2}ight)ight\}\) coincides with \(\mathscr{B}(\mathbb{C}):=\sigma\left(\mathscr{O}_{\mathbb{C}}ight)\).

[ \(\mathscr{O}_{\mathbb{C}}\) is generated by the balls in \(\mathbb{C}\).]

(ii) A map \(h: X ightarrow \mathbb{C}\) is \(\mathscr{A} / \mathscr{C}\)-measurable if, and only if, \(\operatorname{Re} h\) and \(\operatorname{Im} h\) are \(\mathscr{A} / \mathscr{B}(\mathbb{R})\) measurable.

[ the maps \(z \mapsto(\operatorname{Re} z, \operatorname{Im} z)\) and \((x, y) \mapsto z=x+i y\) are continuous.]

A function \(h: E ightarrow \mathbb{C}\) is called \(\mu\)-integrable, if \(\operatorname{Re} h\) and \(\operatorname{Im} h\) are \(\mu\)-integrable; we write \(\mathcal{L}_{\mathbb{C}}^{1}(\mu)\) for the complex \(\mu\)-integrable functions. The integral is extended by linearity, i.e. \(\int h d \mu:=\int \operatorname{Re} h d \mu+i \int \operatorname{Im} h d \mu\). Show the following.

(iii) \(h \mapsto \int h d \mu\) is a \(\mathbb{C}\)-linear map on \(\mathcal{L}_{\mathbb{C}}^{1}(\mu)\).

(iv) \(\operatorname{Re} \int h d \mu=\int \operatorname{Re} h d \mu\) and \(\operatorname{Im} \int h d \mu=\int \operatorname{Im} h d \mu\).

(v) \(\left|\int h d \muight| \leqslant \int|h| d \mu\).

[ since \(\int h d \mu \in \mathbb{C}\) there is some \(\theta \in(-\pi, \pi]\) such that \(e^{i \theta} \int h d \mu \geqslant 0\).]

(vi) \(\mathcal{L}_{\mathbb{C}}^{1}(\mu)=\left\{h: E ightarrow \mathbb{C}: hight.\) is \(\mathscr{A} / \mathscr{C}\)-measurable and \(\left.|h| \in \mathcal{L}_{\mathbb{R}}^{1}(\mu)ight\}\).