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For a nonnegative random variable $X$, Tonelli theorem givesConsequently, for each $a>0$, $\mathbb{E}[X]=\infty$ exactly when ${\sum }_{n\ge 1}\mathbb{P}(X>an)=\infty$; compare the integral on successive intervals of length $a$.

Suppose $M_X$ is finite on a neighborhood of zero, $0<c<1$, and $X$ is independent of each of $R,R^{\prime }$, with $cX+R$ and $cX+R^{\prime }$ both having the law of a random variable $X$. Independence and the Tonelli theorem first give $M_X(t)=M_X(ct)M_R(t)$ as an extended nonnegative integral. Since $0<M_X(ct)<\infty$ and $M_X(t)<\infty$, both remainder moment-generating functions are finite and equal to the displayed ratio on the same neighborhood. The uniqueness theorem for moment-generating functions gives equality of their probability distributions. No independence between $R$ and $R^{\prime }$ is required.

Truncated convolution makes $L^1(0,1)$ a commutative nonunital Banach algebra. The Tonelli theorem gives $\Vert f\ast g{\Vert }_1\le \Vert f{\Vert }_1\Vert g{\Vert }_1$, and the Fubini theorem gives associativity over the integration simplex. Its unitization is $\mathbb{C}\oplus L^1(0,1)$ with norm $|\alpha |+\Vert f{\Vert }_1$ and product $(\alpha ,f)(\beta ,g)=(\alpha \beta ,\alpha g+\beta f+f\ast g)$. The constant function one has convolution powers $t^{n-1}/(n-1)!$, of norm $1/n!$. Thus it is a nonnilpotent quasinilpotent element in this unitization.