$L^p(\mathbb R^n,\mu)$ and $L^p(\mathbb R^n\setminus\{0\},\mu)$

$L^p(\mathbb R^n,\mu)$ and $L^p(\mathbb R^n\setminus\{0\},\mu)$

Let $\mu$ be the Lebesgue measure on $\mathbb R^n$. Some authors usually
use the notation $L^p(\mathbb R^n,\mu)$, that means the set of all
measurable functions $f:\mathbb R^b\to\mathbb C$ so that
$\int\limits_{\mathbb R^n}|f(x)|^pd\mu<\infty$. The same definition for
$L^p(\mathbb R^n\setminus\{0\},\mu)$, the set contains all function
$f:\mathbb R^b\setminus\{0\}\to\mathbb C$ so that $\int\limits_{\mathbb
R^n\setminus\{0\}}|f(x)|^pd\mu<\infty$.
Since $f$ is measurable thus it is no important that if we remove some
point in the domain of function $f$. Is it right? Thus there is no
difference between $L^p(\mathbb R^n,\mu)$ and $L^p(\mathbb
R^n\setminus\{0\},\mu)$. So why the authors still use the second notation?
Or I get some misunderstanding?
As the same if we replace $d\mu$ by $\omega d\mu$, where $\omega$ is a
non-negative, locally integrable function on $\mathbb R^n$. Do they equiv
to each other?