Exercise 1.3.19 (Lebesgue integration is linear)

Let $f,g\in L^1\left({ℝ}^d\to ℂ\right)$ be absolutely integrable Lebesgue measurable functions, and let $c\in ℂ$. We demonstrate the properties of an $\text{∗}$-linear transformation. The steps of this proof will strongly resemble the analogous proof of Exercise 1.3.2.

(i)

(additivity)
First, suppose that $f$ is real-valued. By definition we have $$\begin{array}{llll}\hfill {\int \; <!--\; nolimits\; -->}_{{ℝ}^d}(f+g)& ={\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{(f+g)}_{\text{+}}-{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{(f+g)}_{\text{−}}& \hfill & \\ \multicolumn{4}{c}{\text{Recall that in }\text{Exercise 1.3.2}\text{ we have verified the identity }{(f+g)}_{\text{+}}-{(f+g)}_{\text{−}}=(f_{\text{+}}-f_{\text{−}})+(g_{\text{+}}-g_{\text{−}})\text{. Rewriting it as }{(f+g)}_{\text{+}}+f_{\text{−}}+g_{\text{−}}={(f+g)}_{\text{−}}+f_{\text{+}}+g_{\text{+}}\text{ allows us to conclude that }\int \; <!--\; nolimits\; -->{(f+g)}_{\text{+}}+\int \; <!--\; nolimits\; -->f_{\text{−}}+\int \; <!--\; nolimits\; -->g_{\text{−}}=\int \; <!--\; nolimits\; -->{(f+g)}_{\text{−}}+\int \; <!--\; nolimits\; -->f_{\text{+}}+\int \; <!--\; nolimits\; -->g_{\text{+}}\text{ by Corollary 1.3.14.}}\\ \\ \hfill & ={\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{(f+g)}_{\text{−}}+{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{f}_{\text{+}}+{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{g}_{\text{+}}-{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{f}_{\text{−}}-{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{g}_{\text{−}}-{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{(f+g)}_{\text{−}}& \hfill & \\ \hfill & =\left[{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{f}_{\text{+}}-{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{f}_{\text{−}}\right]+\left[{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{g}_{\text{+}}-{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{g}_{\text{−}}\right]& \hfill & \\ \hfill & ={\int \; <!--\; nolimits\; -->}_{{ℝ}^d}f+{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}g& \hfill & \end{array}$$

Now suppose that $f$ is complex valued. We then have

$$\begin{array}{llll}\hfill {\int \; <!--\; nolimits\; -->}_{{ℝ}^d}(f+g)& ={\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Re (f+g)+i{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Im (f+g)& \hfill & \\ \hfill & ={\int \; <!--\; nolimits\; -->}_{{ℝ}^d}(\mathit{\Re f}+\mathit{\Re g})+i{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}(\mathit{\Im f}+\mathit{\Im g})& \hfill & \\ \hfill & \stackrel{\text{∗}}{\text{=}}{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\mathit{\Re f}+{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\mathit{\Re g}+i{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\mathit{\Im f}+i{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\mathit{\Im g}& \hfill & \\ \hfill & =\left[{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\mathit{\Re f}+i{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\mathit{\Im f}\right]+\left[{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\mathit{\Re g}+i{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\mathit{\Im g}\right]& \hfill & \\ \hfill & ={\int \; <!--\; nolimits\; -->}_{{ℝ}^d}f+{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}g& \hfill & \end{array}$$

(ii)

(homogeneity)
As usual, first suppose that $f$ is real valued, and $c\in ℝ$. We then have $$\begin{array}{llll}\hfill {\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\mathit{cf}& ={\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{(\mathit{cf})}_{\text{+}}-{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{(\mathit{cf})}_{\text{−}}& \hfill & \\ \multicolumn{4}{c}{\text{We have two cases: either }c\ge 0\text{; in which case we one can verify case-wise the identity }{(\mathit{cf})}_{\text{+}}=c{(f)}_{\text{+}}\text{ and }{(\mathit{cf})}_{\text{−}}=c{(f)}_{\text{−}}\text{; and by }\text{Exercise 1.3.10}\text{ (iii) we obtain}}\\ \\ \hfill & ={\int \; <!--\; nolimits\; -->}_{{ℝ}^d}c{(f)}_{\text{+}}-{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}c{(f)}_{\text{−}}=c\cdot \left[{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{f}_{\text{+}}-{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{f}_{\text{−}}\right]=c{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}f& \hfill & \\ \multicolumn{4}{c}{\text{Or we have }c<0\text{ in which case it is verifyable that }{(\mathit{cf})}_{\text{+}}=-c{(f)}_{\text{−}}\text{ and }{(\mathit{cf})}_{\text{−}}=-c{(f)}_{\text{+}}}\\ \\ \hfill & ={\int \; <!--\; nolimits\; -->}_{{ℝ}^d}-c{(f)}_{\text{−}}-{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}-c{(f)}_{\text{+}}=-c\cdot \left[{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{f}_{\text{−}}-{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}{f}_{\text{+}}\right]=c{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}f& \hfill & \end{array}$$

Now suppose that $f$ is complex valued. The scalar multiplication property is demonstrated as follows. We can rewrite $\mathit{cf}$ as

$$\mathit{cf}(x)=[\Re (c)+\mathit{i\Im }(c)]\cdot [\Re (f(x))+\mathit{i\Im }(f(x))]=\Re (c)\cdot \Re (f(x))-\Im (c)\Im (f)+\mathit{i\Im }(c)\Re (f)+\mathit{i\Re }(c)\Im (f)$$

Using this identity we obtain

$$\begin{array}{llll}\hfill {\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\mathit{cf}& ={\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Re (\mathit{cf})+i\cdot {\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Im (\mathit{cf})& \hfill & \\ \hfill & ={\int \; <!--\; nolimits\; -->}_{{ℝ}^d}[\Re (c)\Re (f)-\Im (c)\Im (f)]+i\cdot {\int \; <!--\; nolimits\; -->}_{{ℝ}^d}[\Im (c)\Re (f)+\Re (c)\Im (f)]& \hfill & \\ \hfill & =\Re (c){\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Re (f)-\Im (c){\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Im (f)+\mathit{i\Im }(c)\cdot {\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Re (f)+\Re (c){\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Im (f)& \hfill & \\ \hfill & ={\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Re (f)\cdot \left[\Re (c)+\mathit{i\Im }(c)\right]+i{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Im (f)\left[\Re (c)-1∕\mathit{i\Im }(c)\right]& \hfill & \\ \hfill & =\left[\Re (c)+\mathit{i\Im }(c)\right]\cdot \left({\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Re (f)+i{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Im (f)\right)& \hfill & \\ \hfill & =c\cdot {\int \; <!--\; nolimits\; -->}_{{ℝ}^d}f& \hfill & \end{array}$$

(iii)

(conjuctivity)
The conjuctive property follows as follows $$\begin{array}{llll}\hfill {\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\overline{f}& ={\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Re (\overline{f})+i\cdot {\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Im (\overline{f})& \hfill & \\ \hfill & ={\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Im (f)+i\cdot {\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Re (f)& \hfill & \\ \hfill & =\overline{{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Re (f)+i\cdot {\int \; <!--\; nolimits\; -->}_{{ℝ}^d}\Im (f)}& \hfill & \\ \hfill & =\overline{{\int \; <!--\; nolimits\; -->}_{{ℝ}^d}f}& \hfill & \end{array}$$