0
$\begingroup$

I couldn't find it on Google. I know that $L^1_{\text{loc}}(\mathbb{R}^n) $ is the space of locally integrable functions, but what about $L^\infty_{\text{loc}}(\mathbb{R}^n) $?

Thanks a lot!

  • 2
    $f \in L^{\infty}_{\mathrm{loc}}(\mathbb{R}^n)$ if, for every compact set $K \subset \mathbb{R}^n$, $f \in L^{\infty}(K)$.2012-11-21

1 Answers 1

1

In general, $f\in L^p_{\text{loc}}(\mathbb{R}^n)$ if for every compact $K\subset\mathbb{R}^n$, $f\chi_K\in L^p(\mathbb{R}^n)$ where $\chi_K$ denotes the indicator function of $K$.

  • 0
    Note the minor edit - $f\vert_K$ didn't quite make sense in this context.2012-11-21