Intersection of $L^{p}$ spaces on two sets

Question

If I got a two subsets $\Omega_1$,$\Omega_2$, how can I prove that $L^{p}(\Omega_1)\cap L^{p}(\Omega_2)=L^{p}(\Omega_1\cap \Omega_1)$, $p\in[1,\infty]$. I got problem with norm of $L^{p}(\Omega_1)\cap L^{p}(\Omega_2)$ / how can I define it ? Thanks

Answer 1

Is that true? For $\Omega_{1}={(0,\infty)}$ and $\Omega_{2}=(0,1)$ then $f(x)=1/\sqrt{x}$, $f\in L^{1}(\Omega_{1}\cap\Omega_{2})$ but $f\notin L^{1}(\Omega_{1})$.