I have recently learned that if a function $f:\left[a,b\right]\to\mathbb{R}$ is reimann integrable then it is measurable as a function from $\left(\mathbb{R},\mathcal{L}\right)$ to $\left(\mathbb{R},Borel\left(\mathbb{R}\right)\right)$. This theorem stops being true if we look at f as a function from $\left(\mathbb{R},Borel\left(\mathbb{R}\right)\right)$ to $\left(\mathbb{R},Borel\left(\mathbb{R}\right)\right)$.
My question is if the theorem remains true if you look at f as a function from $\left(\mathbb{R},\mathcal{L}\right)$ to $\left(\mathbb{R},\mathcal{L}\right)$