Let \(f\in L^1_{\text{loc}}(\Omega)\) and \(f_\varepsilon\) its mollification . Then \(f_\varepsilon \to f\) almost everywhere. [1, C.4 Theorem 6]

Proof
This is an consequence of the Lebesgue’s differentiation theorem .

References Link to heading

  1. L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.