The Integral of 1/(1+s²) over ℝ

In the setting of Lebesgue Measure on the Real Line: Standing Notation, let $\pi$ be the number of The Number Pi (i), and let $k:\mathbb{R}\to\mathbb{R}$ be the function $k(s)=1/(1+s^{2})$ (defined since $1+s^{2}\ge1$).

(i) The integral. $k\in\mathcal{M}^{+}(\mathbb{R},\mathcal{B}(\mathbb{R}))$, and its integral is

$$\int_{\mathbb{R}}\frac{1}{1+s^{2}}\,d\lambda(s)=\pi.$$

(ii) Integrability. $k\in\mathcal{L}^{1}(\lambda)$.