The Value of the Gaussian Integral

In the setting of Measurable Functions and the Lebesgue Integral: Notation, let $\lambda$ be Lebesgue measure on $\mathbb{R}$, $\exp$ the exponential function, $\pi$ the number of The Number Pi (i), and $\sqrt{2\pi}$ the square root of $2\pi>0$ (Quarter-Turn Identities and Periodicity of Sine and Cosine (i)). Let $c_{\mathrm{G}}$ be the Gaussian constant of The Gaussian Integral Is Finite and Positive, and the First Two Gaussian Moments.

(i) The value of the Gaussian integral.

$$c_{\mathrm{G}}=\int_{\mathbb{R}}\exp(-x^{2}/2)\,d\lambda(x)=\sqrt{2\pi}.$$