Leave $ f in L ^ 1 ( mathbb R). $ Leave $ r $ be any real number and leave $ B_r: = {x: | x | le r }. $

Then we know $$ int _ { mathbb {R} setminus B_r} | f (x) | dx to0 , text {as} , r to 0. $$

My question is, how to prove this by definition (that is, without appealing to any theorem or elegant things)?

**My method (which involves the dominated convergence theorem):**

$ int _ { mathbb R} | f chi_ {B_r} | le int _ { mathbb R} | f | lt infty $ Y $ f chi_ {B_r} a f $ promptly as $ r to infty $, where $ chi_ {B_r} $ is the characteristic function in $ B_r. $ Then, by the dominated convergence theorem, we are done.