$f \in L^1 (\mathbf{R}) \iff\mathop{\lim_{a \to -\infty}}_{ b \to +\infty} \int_a^b |f(x)| \mathrm{d} x \text{ exists and it is finite}$

Question

Let $(\mathbf{R}, M, m)$ be Lebesgue measure space and $f: \mathbf{R} \rightarrow \mathbf{R}$ be a continuous function. Show that

$$f \in L^1_m (\mathbf{R}) \text{ if and only if } \mathop{\lim_{a \to -\infty}}_{ b \to +\infty} \int_a^b |f(x)| \,\mathrm{d} x \text{ exists and it is finite.}$$

Moreover, $$\int_\mathbf{R} |f(x)| \,\mathrm{d} m = \mathop{\lim_{a \to -\infty}}_{ b \to +\infty} \int_a^b |f(x)| \,\mathrm{d} x \text{ and } \int_\mathbf{R} f(x) \,\mathrm{d} m = \mathop{\lim_{a \to -\infty}}_{ b \to +\infty} \int_a^b f(x) \,\mathrm{d} x.$$

Answer 1

$\Longrightarrow:$ Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a continuous function. Assume that $f \in L^1 (\mathbf{R})$. Then $\int_\mathbf{R} |f| dm < \infty$ by definition. Also we know that every continuous function on the interval $[a,b]$ is Riemann integrable. Hence since $|f|$ is continuous on $[a,b]$, $|f|$ is Riemann integrable on the interval $[a,b]$. We also know that $$\int_{[a,b]} |f| dm = \int_a^b |f| dx.$$

Also since $|f|$ is non-negative, we obtain the following the inequality

$$\int_{[a,b]} |f| dm \leq \int_\mathbf{R} |f| dm \text{ for any } a,b \in \mathbf{R}.$$

Note that the value of the integral is increasing. Therefore by Monotone Convergence Theorem, we have:

$$\lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_a^b |f(x)| dx=\lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_{[a,b]} |f(x)| dm = \int_\mathbf{R} |f| dm.$$

$\Longleftarrow:$ On the other hand, suppose that the limit $\lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_a^b |f(x)| dx$ exists and it is finite. We know that $$\int_{[a,b]} |f| dm = \int_a^b |f| dx.$$

Taking the limit of both sides, we obtain

$$\lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_a^b |f(x)| dx=\lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_{[a,b]} |f(x)| dm= \int_\mathbf{R} |f| dm,$$

as desired.

In order to prove, $\int_\mathbf{R} f(x) dm = \lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_a^b f(x) dx$, we write $f$ as a difference of two non-negative functions $f=f^+ - f^-$. Since $f^+$ and $f^-$ are both non-negative, the result we proved holds. Hence $$\int_\mathbf{R} f^+(x) dm = \lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_a^b f^+(x) dx$$ and $$\int_\mathbf{R} f^-(x) dm = \lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_a^b f^-(x) dx$$ and therefore we conclude that $$\int_\mathbf{R} f(x) dm = \lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_a^b f^+(x) - f^-(x) dx,$$ as desired.