Let \(\Omega\subset \mathbb{R}^n\) be a domain. A function \(v\colon \Omega\to \mathbb{R}\) is called locally integrable if every restriction on a compact subset is integrable. We denote the space of all locally integrable functions on \(\Omega\) with \(L^1_{\text{loc}}(\Omega)\).
Remarks
- Local integrable functions are distributions.
- It is customary to define \(L^p_{\text{loc}}\) for \(p\in [1,\infty)\) in the same manner.
- Convergence in \(L^p_{\text{loc}}(\Omega)\) is given by \(L^p\)-convergence on every compact subset.