Interchanging a limit and an integral: necessary and sufficient conditions

Abstract

Let $\{f_{n}\}_{n \in \mathbb {N}}$ { f n } n ∈ N be a sequence of integrable functions on a σ-finite measure space $(\Omega, \mathscr {F}, \mu )$ ( Ω , F , μ ) . Suppose that the pointwise limit $\lim_{n \uparrow \infty } f_{n}$ lim n ↑ ∞ f n exists μ-a.e. and is integrable. In this setting we provide necessary and sufficient conditions for the following equality to hold: $$ \lim_{n \uparrow \infty } \int f_{n} \, d\mu = \int \lim_{n \uparrow \infty } f_{n} \, d\mu. $$ lim n ↑ ∞ ∫ f n d μ = ∫ lim n ↑ ∞ f n d μ .

Physics-Consistency Condition for Infinite Neural Networks and Experimental Characterization

“…where we used the linearity of the expectation and p(θ) is the probability density function of the weights and biases θ = {v, w, b}. Note that we exchanged limit and integration, which is (typically) permissible if convergence is uniform and the limit function is integrable [44]. This integral can be solved numerically and, in some cases, analytically [28].…”

“…Substituting the NN-GP consistency condition Equation ( 7) into the physics-consistency condition for the GP Equation (4) (or variants thereof) allows us to formulate a physics-consistency condition for the NN directly. Reciting uniform convergence and integrable limits as above used for Equation ( 7) [44], we switch the order of limit, expectation and differentiation:…”

A new stochastic method based on the Taylor expansion to compute response probability densities of uncertain systems

“…To this aim, we can use several theorems concerning the commutation of the limit and the integral. 23 In particular, we apply Theorem 3 given in the following where the uniform convergence is needed but other results, such as the bounded convergence theorem, can be considered. In the bounded convergence theorem, point-wise convergence is needed but each term of the Taylor series shall be bounded.…”

“…The solution of Equation (34) for the rth mode can be expressed by the same form as an SDOF system, see Equation (23).…”

“…Consider the random space $$$ {L}^2\left(\Omega \right) $$$ defined by Now, we start proving that the Taylor expansion is mean square convergent to $$$ g\left(x\left(\omega \right)\right) $$$. To this aim, we can use several theorems concerning the commutation of the limit and the integral 23 . In particular, we apply Theorem 3 given in the following where the uniform convergence is needed but other results, such as the bounded convergence theorem, can be considered.…”