Let $f$ : $\mathbb{R}^d \to \mathbb{C}$ be an absolutely integrable function, and let ε > 0. Show that there exists a ball $B(0, R)$ outside of which f has an $L^1$ norm of at most ε, or in other words that $\int_{\mathbb{R}^d \setminus B(0,R)} |f(x)| dx ≤ ε$.
How should I begin this proof? I have tried to research this but there isn't much literature on this specific property of absolutely integrable functions.
Partition $\mathbb{R}^d$ into $B_n = \{x\in\mathbb{R}^d : n\leq \|x\| < n+1 \}$, for $n=0,1,\ldots$
Let $b_n = \int_{B_N} |f|$ so that $\int |f| = \sum_{n=0}^\infty b_n$.
(Here I am using the dominated convergence theorem.)
Since the sum is finite, for sufficiently large $R$ we have $\sum_{n=R}^\infty b_n < \epsilon$.