If a function $f : \mathbb{R}^d \to [-\infty,\infty]$ is Lebesgue integrable, then by definition we have $$\int_{\mathbb{R}^d} |f(x)| \, dx < +\infty.$$ Is it possible to say that there exists a nonnegative $g(x)$ such that $$\int_{\mathbb{R}^d} |f(x)| \, dx \le \int_{\mathbb{R}^d} g(x) \, dx < +\infty,$$ and so $|f(x)| < g(x)$ for all $x \in \mathbb{R}^d$?
The reason I ask is because I want to use the Dominated Convergence Theorem for another exercise I am working on.
Counterexample: Let $c=\int_{\mathbb{R}^d} |f(x)|dx$. Then $\int_{\mathbb{R}^d}|f(x)|\leq \int_{\mathbb{R}^d} c\chi_{I^d}dx<\infty$ but it is not true that $|f(x)|<c\chi_{I^d}$ for all $x$. Here $I^d=[0,1]^d$