Exercise 1.3.20 (Basic properties of the Lebesgue integral of the absolutely integrable functions)

Question

Let $f,g:\mathbb{R}^d	o\mathbb{C}$ be absolutely integrable Lebesgue measurable functions. Show that
\begin{enumerate}
\item (translation invariance) for $f'(x):=f(x+y)$, $y\in\mathbb{R}^d$ we have $\int_{\mathbb{R}^d} f' = \int_{\mathbb{R}^d} f$.
\item (linear change of variables) if $T:\mathbb{R}^d	o\mathbb{R}^d$ is an invertible linear transformation then $\int_{\mathbb{R}^d} f(T) = \frac{1}{|\det T|} \int_{\mathbb{R}^d} f$.
\item (compatibility with the Riemann integral) if $f$ is also (extended via zero support complement) Riemann integrable, then $\mathrm{Riemann}\int_{[a,b]} f = \int_{\mathbb{R}^d} f$
\end{enumerate}

Elu Thingol · Accepted Answer

\begin{enumerate}[label=(oman*)]
	\item By definition we have
	\begin{align*}
		\int_{\mathbb{R}^d} f &= \int_{\mathbb{R}^d} \Re f + i \int_{\mathbb{R}^d} \Im f\
		&= \int_{\mathbb{R}^d} (\Re f)_+ - \int_{\mathbb{R}^d} (\Re f)_- + i\int_{\mathbb{R}^d} (\Im f)_+ - i \int_{\mathbb{R}^d} (\Im f)_-
		\end{align*}
		Define $(\Re f)'_+(x) := (\Re f)_+(x+y)$, and similarly $(\Re f)'_-(x) := (\Re f)_-(x+y)$, $(\Im f)'_+(x) := (\Re f)_+(x+y)$, $(\Im f)'_-(x) := (\Im f)_-(x+y)$. Since all of them are unsigned measurable functions, we can apply \href{./exercise_1-3-15}{Exercise 1.3.15} to obtain
		\begin{align*}
		&= \int_{\mathbb{R}^d} (\Re f)'_+ - \int_{\mathbb{R}^d} (\Re f)'_- + i\int_{\mathbb{R}^d} (\Im f)'_+ - i \int_{\mathbb{R}^d} (\Im f)'_-\
		&= \int_{\mathbb{R}^d} (\Re f)'  + i\int_{\mathbb{R}^d} (\Im f)'
		\end{align*}
		Obviously, for any $x \in \mathbb{R}^d$ we have $(\Re f)'(x) + i (\Im f)'(x) = (\Re f)(x+y) + i(\Im f)(x+y) = f(x+y) = f'(x)$. Thus,
		\begin{align*}
		&= \int_{\mathbb{R}^d} f'
		\end{align*}
		
		\item NOT SOLVED
		
		\item In this case we assume that $f$ is real-valued. We start with the Lebesgue integral:
		\begin{align*}
		\int_{\mathbb{R}^d} f &= \int_{\mathbb{R}^d} f_+ + \int_{\mathbb{R}^d} f_-\
		\end{align*}
		By \href{./exercise_1-3-13}{Exercise 1.3.13} the Lebesgue integral of the unsigned Lebesgue measurable function is equal to the area under its curve. In other words, define $E_+:= \{(x,t)\in\mathbb{R}^d	imes [0,+\infty]: 0 \leq t \leq f(x)\} = \{(x,t)\in\mathbb{R}^d	imes [0,+\infty]: 0 \leq t \leq f_+(x)\}$ and $E_-=\{(x,t)\in\mathbb{R}^d	imes [-\infty,0]: f(x) \leq t \leq 0\} \sim \{(x,t)\in\mathbb{R}^d	imes [0,+\infty]: 0 \leq t \leq f_-(x)\}$. We then have
		\begin{align*}
		&= m(E_+) - m(E_-)
		\end{align*}
		Using \href{./exercise_1-1-25}{Exercise 1.1.25} we see that
		\begin{align*}
		&= \mathrm{Riemann}\int_{[a,b]} f.
		\end{align*}
\end{enumerate}

Exercise 1.3.20 (Basic properties of the Lebesgue integral of the absolutely integrable functions)

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Exercise 1.3.20 (Basic properties of the Lebesgue integral of the absolutely integrable functions)

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