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If the variables are independent random variables, apply the defining product rule first to indicator functions, then to simple functions, and finally use bounded measurable approximation to obtain the displayed identity. Conversely, choosing $f_n=1_{A_n}$ giveswhich is independence.

Let $S_m={\sum }_{n\le m}{X}_n$. Independence and zero means imply orthogonality in $L^2$, soIf the variance series converges, $(S_m)$ is Cauchy in the complete space $L^2$ and hence converges there. Conversely, $L^2$ convergence makes $(S_m)$ Cauchy, so every tail of the nonnegative variance series tends to zero; therefore ${\sum }_n{\sigma }_n^2<\infty$.

A point $y$ is a regular value of $f$ when every derivative $D_{x}f$ for $x$ in $f^{-1}(y)$ is surjective. Sard theorem says that the critical values of a smooth map have measure zero. The stack-of-records theorem says that the inverse image of a regular value is a smooth submanifold of codimension $\dim Y$.

Maps $f$ and $g$ are smoothly homotopic when a smooth $F:X\times [0,1]\to Y$ has endpoint maps $f$ and $g$. Perturb such a homotopy relative to its ends so that it is transverse to $y$. Then $F^{-1}(y)$ is a compact one-dimensional manifold whose boundary is $f^{-1}(y)\bigsqcup g^{-1}(y)$. Every compact one-dimensional manifold has an even number of boundary points, proving the parity formula.

The hypothesis on $U$ says that the degree modulo two of $h$ is one. Suppose no $h(x)$ and $h(-x)$ were antipodal. Thenwould be defined, even, and homotopic to $h$ by normalized straight-line interpolation. For a regular value, the fibres of the even map $k$ occur in antipodal pairs, so its degree modulo two is zero. Homotopy invariance gives a contradiction. Hence some antipodal pair is mapped to an antipodal pair.