I have an observation in my real analysis lecture notes that states that if $f,g:\,(X,\mu)\to\mathbb{R}$ (with Borel's -$\sigma$ algebra ) and $f=g$ almost everywhere then if $g$ is Lebesgue measurable then so if $f$.
I don't understand why this is true, can someone please explain ?
I tried looking at some Borel set and on it source, but I can't figure why if we change $g$ in some measure $0$ of points then the source is still in Lebesgue -$\sigma$ algebra