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Suppose an unknown has prior distribution $\Pi$ and its conditional observation law $P_u$ has jointly measurable Radon-Nikodym derivative $L(u,m)$ relative to a fixed observation law $P_0$. If $L>0$ for $\Pi (du)P_0(dm)$-almost every pair, thendefines the posterior distribution for almost every observation. Indeed, $\int L(u,m)P_0(dm)=1$ for almost every $u$, so Tonelli theorem gives $\int Z(m)P_0(dm)=1$. Hence $Z$ is finite almost everywhere, and positivity of $L$ and Fubini's theorem give $Z>0$ almost everywhere. The data law is $Z{P}_0$, and integration of the proposed posterior against this data law recovers the joint law. This proves the conditional distribution property and also shows that the data law is an equivalent probability measure to $P_0$.

The zero set $\{t\ge 0:B_t=0\}$ of one-dimensional Brownian motion is almost surely closed, uncountable, and has zero Lebesgue measure. The last fact follows from Tonelli theorem because $\mathbb{P}(B_t=0)=0$ for every $t>0$.

For independent integrable variables, $\mathbb{E}{\prod }_i{X}_i={\prod }_i\mathbb{E}{X}_i$. Tonelli theorem first gives $\mathbb{E}{\prod }_i|X_i|={\prod }_i\mathbb{E}|X_i|<\infty$; then Fubini's theorem factors the signed joint integral.

Let $X_n$ be integrable independent and identically distributed random variables, and let $T\ge 1$ be a stopping time for their natural filtration, with $\mathbb{E}T<\infty$. Since $\{T\ge n\}$ is measurable before $X_n$ is observed, independence and the Tonelli theorem give $\mathbb{E}|X_T|\le {\sum }_n\mathbb{E}[|X_n|1_{\{T\ge n\}}]=\mathbb{E}|X_1|\mathbb{E}T$. The event $T=n$ need not be independent of $X_n$, so the selected increment can be biased.

The integrated mean squared error is $\mathbb{E}{\int }_D(\hat{m}(x)-m(x){)}^{2}dx$ over a specified domain $D$. The Tonelli theorem permits integration of the pointwise bias-variance decomposition of mean squared error.

For an $L^2$ kernel and an independent sample from a density $f$, the centred kernel density estimator hasIndependence removes cross terms and Tonelli theorem integrates the pointwise variance. Young's convolution inequality ensures $K_h\ast f\in L^2$ even without $f\in L^2$. The Cauchy-Schwarz inequality yields the expected-norm bound $\mathbb{E}\Vert {\hat{f}}_{n,h}-K_h\ast f{\Vert }_2\le \Vert K{\Vert }_2/\sqrt{nh}$.

For a nonnegative measurable function $g$,On a graph, applying this identity to $|g(x)-g(y)|$ expresses its total variation as the integral of the edge boundaries of the superlevel sets.

More generally, if $\phi :[0,\infty )\to [0,\infty )$ is differentiable, increasing, and satisfies $\phi (0)=0$, then Tonelli theorem and the fundamental theorem of calculus give

For every nonprincipal Dirichlet character, $L(1,\chi )\ne 0$. A nonreal character is covered by Euler product positivity for L-function nonvanishing at height zero, since its squared character is nonprincipal.

For a real character, suppose $L(1,\chi )=0$. The simple zeta pole would cancel, making $F(s)=\zeta (s)L(s,\chi )$ an entire function. Its nonnegative zeta-times-real-L coefficients satisfy $a_n\ge 0$ and $a_{m^2}\ge 1$. Termwise derivatives at two give $(-1{)}^k{F}^{(k)}(2)={\sum }_n{a}_n(\log n{)}^k{n}^{-2}$. The Taylor series of this entire function at two converges at zero. Every term there is nonnegative, and Tonelli theorem identifies its sum asThe last series diverges because all square coefficients are at least one. This contradiction proves the claim. It is the positive-coefficient mechanism behind the more general Landau theorem for a Dirichlet series with nonnegative coefficients.