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# A characterization of the Borel $\sigma$-algebra in $\mathbb{R}^n$

Consider the linear spacet $\mathcal{F}(\mathbb{R}^n)$ of all real functions defined in $\mathbb{R}^n$. It is well known that the subspace $\mathcal{C}(\mathbb{R}^n)$ of all real valued continuous function defined in $\mathbb{R}^n$ is stable with respect to the uniform (convergence) limit of elements in $\mathcal{C}(\mathbb{R}^n)$.

Question 1: Which is the smallest set (with respect to inclusion relation) containing $\mathcal{C}(\mathbb{R}^n)$ and stable with respect to pointwise convergence?

Question 2: Which is the smallest linear subspace of $\mathcal{F}(\mathbb{R}^n)$ which is stable with respect to pointwise convergence?