However, $\exists f:\mathbb{R}\to \mathbb{R}$ continuous such that $$f:(\mathbb{R},\mathcal{M}(\lambda^{*}))\to(\mathbb{R},\mathcal{M}(\lambda^{*}))$$ is not a measurable function. This is our prime motivator for restricting ourselves to the Borel σ-algebra instead of Lebesgue Measurable σ-algebra.