# real analysis – Integral of the function L ^ 1 out of a large ball

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.