This is a question from Terrence Tao's textbook "Introduction to Measure Theory" Let f : $\mathbb{R}$ → $\mathbb{C}$ be a measurable function supported on a set of finite measure, and let ε > 0. Show that there exists a measurable set E ⊂ $\mathbb{R}^d$ of measure at most ε outside of which f is locally bounded, or in other words that for every R > 0 there exists M < ∞ such that |f(x)| ≤ M for all x ∈ B(0,R)\E.
I'm not quite sure where to start here. In the text description, it says that this is a "littlewood-like" principle but not sure of the relation. Could I get a hint towards this?
Begin by finding a set $ E_1 $ of measure less than $ \varepsilon / 2 $ outside of which $ f $ is bounded on $ B(0,1)\backslash E_1 $. Then find a set $ E_2 $ of measure less than $ \varepsilon / 4 $ outside of which $ f $ is bounded on $ B(0,2)\backslash E_2 $. Keep going, and define $ E_1, E_2, E_3, \ldots $. Does this hint help? Do you see how you can proceed from here?