measure theory – Inclusions between $L^p$ spaces.

Theorem. Let $mu(X)<infty$. Then $$1le p le qleinftyimplies L^q(X,mathcal{A},mu)subseteq L^p(X,mathcal{A},mu) $$

Def. Let $Omegasubseteqmathbb{R}^n$ be an open subset, $fcolonOmegato (-infty,+infty)$ q.o defined. We said that the function $f$ is locally integrable in $Omega$ if $fin L^1(G,mathcal{L}(mathbb{R}^n)cap G,lambda)$ for all $Ginmathcal{L}(mathbb{R^n})$ such that $overline{G}subseteqOmega.$

In the definition $lambda$ is the Lebesgue measure on $mathbb{R}^n$.

We denote the set of locally integrable function with $L^1_{text{loc}}$

I must prove that $$L^p(Omega)subseteq L^1_{text{loc}}(Omega)quadtext{for all};pin(1,+infty)$$

using the previous theorem.

Naturally $$L^1(Omega)subseteq L^1_{text{loc}}(Omega)$$

Now I don’t know how to proceed. Could anyone give me a suggestion? Thanks!