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My answer is inspired by the one of coudy: how many scientists who deal with the Lebesgue integral on a daily basis know that there exists a necessary and sufficient condition for the passage to the limit under the integral symbol? I learned about it while reading some papers on the history of Italian mathematics written by the late Gaetano Fichera: the result, stated in modern language ([4], Ch. VIII, pp. 110-128), is reported below.

Definition 1. Let $(E,\mathcal{E})$ be a measure space and $\phi:\mathcal{E}\to\overline{\mathbb{R}}$ a numerical set function: $\phi$ is called exhaustive if $$ \lim_n\phi(A_n)=0 $$ for all families $\{A_n\}$ of pairwise disjoint sets in $\mathcal{E}$.

Definition 2. Let $(E,\mathcal{E})$ be a measure space and $H$ a set (and thus possibly a family) of numerical set functions defined on $\mathcal{E}$: $H$ is called uniformly exhaustive if the numerical set function $$ A\mapsto\sup_{\phi\in H} \vert\phi(A)\vert\;\text{ is exhaustive.} $$

Cafiero's theorem (on the passage to the limit under the integral). Let $(E,\mathcal{E})$ be a measure space, $(\mu_n)_{n\geq 1}$ be a sequence of real measures and $(f_n)_{n\geq 1}$ be a sequence of real functions such that $f_n\in\mathcal{L}^1(\vert\mu_n\vert)$ for all $n$ (here the notation $\vert\mu\vert$ identifies the variation of the measure $\mu$). Suppose moreover that the following pointwise limits exist \begin{split} \lim_{n\to\infty} \mu_n &=\mu\\ \lim_{n\to\infty} f_n &=f \end{split} where $\mu$ and $f$ are respectively a real measure and a real function. Then $$ \lim_{n\to\infty} \int f_n\mathrm{d}\mu_n = \int f \mathrm{d}\mu\iff\text{$(f_n\cdot\mu_n)_{n\geq 1}$ is uniformly exaustive.} $$

The result was originally proved by Cafiero in [1] (see also book [2], ch. VII, §2 pp. 377-392), who generalized the concept of uniform additivity introduced before and independently by Renato Caccioppoli and Vladimir Dubrovskii: that theorem includes the ones of Nykodym, Vitali, Hahn and Saks and an earlier result of Gaetano Fichera [3], where a necessary and sufficient condition was proved for the integral with respect to a given fixed measure. The work of Cafiero is cited in the bibliography of the treatise on linear operators by Dunford and Schwartz but, to my knowledge, the only English reference discussing (very briefly) his contribution is the recent treatise of Vladimir Bogachev.

[1] Cafiero, F. (1953), "Sul passaggio al limite sotto il segno d'integrale per successioni d'integrali di Stieltjes-Lebesgue negli spazi astratti, con masse variabili con gli integrandi [On the passage to the limit under the sign of integral for sequences of Stieltjes–Lebesgue integrals in abstract spaces, with masses varying jointly with integrands]" (Italian), Rendiconti del Seminario Matematico della Università di Padova, 22: 223–245, MR0057951, Zbl 0052.05003.

[2] Cafiero, F. (1959), Misura e integrazione [Measure and integration] (Italian), Monografie matematiche del Consiglio Nazionale delle Ricerche 5, Roma: Edizioni Cremonese, pp. VII+451, MR0215954, Zbl 0171.01503.

[3] Fichera, G. (1943), "Intorno al passaggio al limite sotto il segno d'integrale" [On the passage to the limit under the sign of integral] (Italian), Portugaliae Mathematica, 4 (1): 1–20, MR0009192, Zbl 0063.01364.

[4] Letta, G. (2013), Argomenti scelti di Teoria della Misura [Selected topics in Measure Theory], (in Italian) Quaderni dell'Unione Matematica Italiana 54, Bologna: Unione Matematica Italiana, pp. XI+183, ISBN 88-371-1880-5, Zbl 1326.28001.

My answer is inspired by the one of coudy: how many scientists who deal with the Lebesgue integral on a daily basis know that there exists a necessary and sufficient condition for the passage to the limit under the integral symbol? I learned about it while reading some papers on the history of Italian mathematics written by the late Gaetano Fichera: the result, stated in modern language ([4], Ch. VIII, pp. 110-128), is reported below.

Definition 1. Let $(E,\mathcal{E})$ be a measure space and $\phi:\mathcal{E}\to\overline{\mathbb{R}}$ a numerical set function: $\phi$ is called exhaustive if $$ \lim_n\phi(A_n)=0 $$ for all families $\{A_n\}$ of pairwise disjoint sets in $\mathcal{E}$.

Definition 2. Let $(E,\mathcal{E})$ be a measure space and $H$ a set (and thus possibly a family) of numerical set functions defined on $\mathcal{E}$: $H$ is called uniformly exhaustive if the numerical set function $$ A\mapsto\sup_{\phi\in H} \vert\phi(A)\vert\;\text{ is exhaustive.} $$

Cafiero's theorem (on the passage to the limit under the integral). Let $(E,\mathcal{E})$ be a measure space, $(\mu_n)_{n\geq 1}$ be a sequence of real measures and $(f_n)_{n\geq 1}$ be a sequence of real functions such that $f_n\in\mathcal{L}^1(\vert\mu_n\vert)$ for all $n$ (here the notation $\vert\mu\vert$ identifies the variation of the measure $\mu$). Suppose moreover that the following pointwise limits exist \begin{split} \lim_{n\to\infty} \mu_n &=\mu\\ \lim_{n\to\infty} f_n &=f \end{split} where $\mu$ and $f$ are respectively a real measure and a real function. Then $$ \lim_{n\to\infty} \int f_n\mathrm{d}\mu_n = \int f \mathrm{d}\mu\iff\text{$(f_n\cdot\mu_n)_{n\geq 1}$ is uniformly exaustive.} $$

The result was originally proved by Cafiero in [1] (see also book [2], ch. VII, §2 pp. 377-392), who generalized the concept of uniform additivity introduced before and independently by Renato Caccioppoli and Vladimir Dubrovskii: that theorem includes the ones of Nykodym, Vitali, Hahn and Saks and an earlier result of Gaetano Fichera [3], where a necessary and sufficient condition was proved for the integral with respect to a given fixed measure. The work of Cafiero is cited in the bibliography of the treatise on linear operators by Dunford and Schwartz but, to my knowledge, the only English reference discussing (very briefly) his contribution is the recent treatise of Vladimir Bogachev.

[1] Cafiero, F. (1953), "Sul passaggio al limite sotto il segno d'integrale per successioni d'integrali di Stieltjes-Lebesgue negli spazi astratti, con masse variabili con gli integrandi [On the passage to the limit under the integral symbol for sequences of Stieltjes–Lebesgue integrals in abstract spaces, with masses varying jointly with integrands]" (Italian), Rendiconti del Seminario Matematico della Università di Padova, 22: 223–245, MR0057951, Zbl 0052.05003.

[2] Cafiero, F. (1959), Misura e integrazione [Measure and integration] (Italian), Monografie matematiche del Consiglio Nazionale delle Ricerche 5, Roma: Edizioni Cremonese, pp. VII+451, MR0215954, Zbl 0171.01503.

[3] Fichera, G. (1943), "Intorno al passaggio al limite sotto il segno d'integrale" [On the passage to the limit under the integral symbol] (Italian), Portugaliae Mathematica, 4 (1): 1–20, MR0009192, Zbl 0063.01364.

[4] Letta, G. (2013), Argomenti scelti di Teoria della Misura [Selected topics in Measure Theory], (in Italian) Quaderni dell'Unione Matematica Italiana 54, Bologna: Unione Matematica Italiana, pp. XI+183, ISBN 88-371-1880-5, Zbl 1326.28001.

My answer is inspired by the one of coudy: how many scientists who deal with the Lebesgue integral on a daily basis know that there exists a necessary and sufficient condition for the passage to the limit under the integral symbol? I learned about it while reading some papers on the history of Italian mathematics written by the late Gaetano Fichera: the result, stated in modern language ([4], Ch. VIII, pp. 110-128), is reported below.

Definition 1. Let $(E,\mathcal{E})$ be a measure space and $\phi:\mathcal{E}\to\overline{\mathbb{R}}$ a numerical set function: $\phi$ is called exhaustive if $$ \lim_n\phi(A_n)=0 $$ for all families $\{A_n\}$ of pairwise disjoint sets in $\mathcal{E}$.

Definition 2. Let $(E,\mathcal{E})$ be a measure space and $H$ a set (and thus possibly a family) of numerical set functions defined on $\mathcal{E}$: $H$ is called uniformly exhaustive if the numerical set function $$ A\mapsto\sup_{\phi\in H} \vert\phi(A)\vert\;\text{ is exhaustive.} $$

Cafiero's theorem (on the passage to the limit under the integral). Let $(E,\mathcal{E})$ be a measure space, $(\mu_n)_{n\geq 1}$ be a sequence of real measures and $(f_n)_{n\geq 1}$ be a sequence of real functions such that $f_n\in\mathcal{L}^1(\vert\mu_n\vert)$ for all $n$ (here the notation $\vert\mu\vert$ identifies the variation of the measure $\mu$). Suppose moreover that the following pointwise limits exist \begin{split} \lim_{n\to\infty} \mu_n &=\mu\\ \lim_{n\to\infty} f_n &=f \end{split} where $\mu$ and $f$ are respectively a real measure and a real function. Then $$ \lim_{n\to\infty} \int f_n\mathrm{d}\mu_n = \int f \mathrm{d}\mu\iff\text{$(f_n\cdot\mu_n)_{n\geq 1}$ is uniformly exaustive.} $$

The result was originally proved by Cafiero in [1] (see also book [2], ch. VII, §2 pp. 377-392), who generalized the concept of uniform additivity introduced before and independently by Renato Caccioppoli and Vladimir Dubrovskii: that theorem includes the ones of Nykodym, Vitali, Hahn and Saks and an earlier result of Gaetano Fichera [3], where a necessary and sufficient condition was proved for the integral with respect to a given fixed measure. The work of Cafiero is cited in the bibliography of the treatise on linear operators by Dunford and Schwartz but, to my knowledge, the only English reference discussing (very briefly) his contribution is the recent treatise of Vladimir Bogachev.

[1] Cafiero, F. (1953), "Sul passaggio al limite sotto il segno d'integrale per successioni d'integrali di Stieltjes-Lebesgue negli spazi astratti, con masse variabili con gli integrandi" [On the passage to the limit under the sign of integral for sequences of Stieltjes–Lebesgue integrals in abstract spaces, with masses varying jointly with integrands], Rendiconti del Seminario Matematico della Università di Padova (in Italian), 22: 223–245, MR 0057951, Zbl 0052.05003.

[2] Cafiero, F. (1959), Misura e integrazione [Measure and integration], (in Italian), Monografie matematiche del Consiglio Nazionale delle Ricerche 5, Roma: Edizioni Cremonese, pp. VII+451, MR 0215954, Zbl 0171.01503.

[3] Fichera, G. (1943), "Intorno al passaggio al limite sotto il segno d'integrale" [On the passage to the limit under the sign of integral], Portugaliae Mathematica (in Italian), 4 (1): 1–20, MR 0009192, Zbl 0063.01364.

[4] Letta, G. (2013), Argomenti scelti di Teoria della Misura [Selected topics in Measure Theory], (in Italian) Quaderni dell'Unione Matematica Italiana 54, Bologna: Unione Matematica Italiana, pp. XI+183, ISBN 88-371-1880-5, Zbl 1326.28001.

My answer is inspired by the one of coudy: how many scientists who deal with the Lebesgue integral on a daily basis know that there exists a necessary and sufficient condition for the passage to the limit under the integral symbol? I learned about it while reading some papers on the history of Italian mathematics written by the late Gaetano Fichera: the result, stated in modern language ([4], Ch. VIII, pp. 110-128), is reported below.

Definition 1. Let $(E,\mathcal{E})$ be a measure space and $\phi:\mathcal{E}\to\overline{\mathbb{R}}$ a numerical set function: $\phi$ is called exhaustive if $$ \lim_n\phi(A_n)=0 $$ for all families $\{A_n\}$ of pairwise disjoint sets in $\mathcal{E}$.

Definition 2. Let $(E,\mathcal{E})$ be a measure space and $H$ a set (and thus possibly a family) of numerical set functions defined on $\mathcal{E}$: $H$ is called uniformly exhaustive if the numerical set function $$ A\mapsto\sup_{\phi\in H} \vert\phi(A)\vert\;\text{ is exhaustive.} $$

Cafiero's theorem (on the passage to the limit under the integral). Let $(E,\mathcal{E})$ be a measure space, $(\mu_n)_{n\geq 1}$ be a sequence of real measures and $(f_n)_{n\geq 1}$ be a sequence of real functions such that $f_n\in\mathcal{L}^1(\vert\mu_n\vert)$ for all $n$ (here the notation $\vert\mu\vert$ identifies the variation of the measure $\mu$). Suppose moreover that the following pointwise limits exist \begin{split} \lim_{n\to\infty} \mu_n &=\mu\\ \lim_{n\to\infty} f_n &=f \end{split} where $\mu$ and $f$ are respectively a real measure and a real function. Then $$ \lim_{n\to\infty} \int f_n\mathrm{d}\mu_n = \int f \mathrm{d}\mu\iff\text{$(f_n\cdot\mu_n)_{n\geq 1}$ is uniformly exaustive.} $$

The result was originally proved by Cafiero in [1] (see also book [2], ch. VII, §2 pp. 377-392), who generalized the concept of uniform additivity introduced before and independently by Renato Caccioppoli and Vladimir Dubrovskii: that theorem includes the ones of Nykodym, Vitali, Hahn and Saks and an earlier result of Gaetano Fichera [3], where a necessary and sufficient condition was proved for the integral with respect to a given fixed measure. The work of Cafiero is cited in the bibliography of the treatise on linear operators by Dunford and Schwartz but, to my knowledge, the only English reference discussing (very briefly) his contribution is the recent treatise of Vladimir Bogachev.

[1] Cafiero, F. (1953), "Sul passaggio al limite sotto il segno d'integrale per successioni d'integrali di Stieltjes-Lebesgue negli spazi astratti, con masse variabili con gli integrandi [On the passage to the limit under the sign of integral for sequences of Stieltjes–Lebesgue integrals in abstract spaces, with masses varying jointly with integrands]" (Italian), Rendiconti del Seminario Matematico della Università di Padova, 22: 223–245, MR0057951, Zbl 0052.05003.

[2] Cafiero, F. (1959), Misura e integrazione [Measure and integration] (Italian), Monografie matematiche del Consiglio Nazionale delle Ricerche 5, Roma: Edizioni Cremonese, pp. VII+451, MR0215954, Zbl 0171.01503.

[3] Fichera, G. (1943), "Intorno al passaggio al limite sotto il segno d'integrale" [On the passage to the limit under the sign of integral] (Italian), Portugaliae Mathematica, 4 (1): 1–20, MR0009192, Zbl 0063.01364.

[4] Letta, G. (2013), Argomenti scelti di Teoria della Misura [Selected topics in Measure Theory], (in Italian) Quaderni dell'Unione Matematica Italiana 54, Bologna: Unione Matematica Italiana, pp. XI+183, ISBN 88-371-1880-5, Zbl 1326.28001.

My answer is inspired by the one of coudy: how many scientists who deal with the Lebesgue integral on a daily basis know that there exists a necessary and sufficient condition for the passage to the limit under the integral symbol? I learned about it while reading some papers on the history of Italian mathematics written by the late Gaetano Fichera: the result, stated in modern language ([4], Ch. VIII, pp. 111-128), is reported below.

Definition 1. Let $(E,\mathcal{E})$ be a measure space and $\phi:\mathcal{E}\to\overline{\mathbb{R}}$ a numerical set function: $\phi$ is called exhaustive if $$ \lim_n\phi(A_n)=0 $$ for all families $\{A_n\}$ of pairwise disjoint sets in $\mathcal{E}$.

Definition 2. Let $(E,\mathcal{E})$ be a measure space and $H$ a set (and thus possibly a family) of numerical set functions defined on $\mathcal{E}$: $H$ is called uniformly exhaustive if the numerical set function $$ A\mapsto\sup_{\phi\in H} \vert\phi(A)\vert\;\text{ is exhaustive.} $$

Cafiero's theorem (on the passage to the limit under the integral). Let $(E,\mathcal{E})$ be a measure space, $(\mu_n)_{n\geq 1}$ be a sequence of real measures and $(f_n)_{n\geq 1}$ be a sequence of real functions such that $f_n\in\mathcal{L}^1(\vert\mu_n\vert)$ for all $n$ (here the notation $\vert\mu\vert$ identifies the variation of the measure $\mu$). Suppose moreover that the following pointwise limits exist \begin{split} \lim_{n\to\infty} \mu_n &=\mu\\ \lim_{n\to\infty} f_n &=f \end{split} where $\mu$ and $f$ are respectively a real measure and a real function. Then $$ \lim_{n\to\infty} \int f_n\mathrm{d}\mu_n = \int f \mathrm{d}\mu\iff\text{$(f_n\cdot\mu_n)_{n\geq 1}$ is uniformly exaustive.} $$

The result was originally proved by Cafiero in [1] (see also book [2], ch. VII, §2 pp. 377-392), who generalized the concept of uniform additivity introduced before and independently by Renato Caccioppoli and Vladimir Dubrovskii: that theorem includes the ones of Nykodym, Vitali, Hahn and Saks and an earlier result of Gaetano Fichera [3], where a necessary and sufficient condition was proved for the integral with respect to a given fixed measure. The work of Cafiero is cited in the bibliography of the treatise on linear operators by Dunford and Schwartz but, to my knowledge, the only English reference discussing (very briefly) his contribution is the recent treatise of Vladimir Bogachev.

[1] Cafiero, F. (1953), "Sul passaggio al limite sotto il segno d'integrale per successioni d'integrali di Stieltjes-Lebesgue negli spazi astratti, con masse variabili con gli integrandi" [On the passage to the limit under the sign of integral for sequences of Stieltjes–Lebesgue integrals in abstract spaces, with masses varying jointly with integrands], Rendiconti del Seminario Matematico della Università di Padova (in Italian), 22: 223–245, MR 0057951, Zbl 0052.05003.

[2] Cafiero, F. (1959), Misura e integrazione [Measure and integration], (in Italian), Monografie matematiche del Consiglio Nazionale delle Ricerche 5, Roma: Edizioni Cremonese, pp. VII+451, MR 0215954, Zbl 0171.01503.

[3] Fichera, G. (1943), "Intorno al passaggio al limite sotto il segno d'integrale" [On the passage to the limit under the sign of integral], Portugaliae Mathematica (in Italian), 4 (1): 1–20, MR 0009192, Zbl 0063.01364.

[4] Letta, G. (2013), Argomenti scelti di Teoria della Misura [Selected topics in Measure Theory], (in Italian) Quaderni dell'Unione Matematica Italiana 54, Bologna: Unione Matematica Italiana, pp. XI+183, ISBN 88-371-1880-5, Zbl 1326.28001.

My answer is inspired by the one of coudy: how many scientists who deal with the Lebesgue integral on a daily basis know that there exists a necessary and sufficient condition for the passage to the limit under the integral symbol? I learned about it while reading some papers on the history of Italian mathematics written by the late Gaetano Fichera: the result, stated in modern language ([4], Ch. VIII, pp. 110-128), is reported below.

Definition 1. Let $(E,\mathcal{E})$ be a measure space and $\phi:\mathcal{E}\to\overline{\mathbb{R}}$ a numerical set function: $\phi$ is called exhaustive if $$ \lim_n\phi(A_n)=0 $$ for all families $\{A_n\}$ of pairwise disjoint sets in $\mathcal{E}$.

Definition 2. Let $(E,\mathcal{E})$ be a measure space and $H$ a set (and thus possibly a family) of numerical set functions defined on $\mathcal{E}$: $H$ is called uniformly exhaustive if the numerical set function $$ A\mapsto\sup_{\phi\in H} \vert\phi(A)\vert\;\text{ is exhaustive.} $$

Cafiero's theorem (on the passage to the limit under the integral). Let $(E,\mathcal{E})$ be a measure space, $(\mu_n)_{n\geq 1}$ be a sequence of real measures and $(f_n)_{n\geq 1}$ be a sequence of real functions such that $f_n\in\mathcal{L}^1(\vert\mu_n\vert)$ for all $n$ (here the notation $\vert\mu\vert$ identifies the variation of the measure $\mu$). Suppose moreover that the following pointwise limits exist \begin{split} \lim_{n\to\infty} \mu_n &=\mu\\ \lim_{n\to\infty} f_n &=f \end{split} where $\mu$ and $f$ are respectively a real measure and a real function. Then $$ \lim_{n\to\infty} \int f_n\mathrm{d}\mu_n = \int f \mathrm{d}\mu\iff\text{$(f_n\cdot\mu_n)_{n\geq 1}$ is uniformly exaustive.} $$

The result was originally proved by Cafiero in [1] (see also book [2], ch. VII, §2 pp. 377-392), who generalized the concept of uniform additivity introduced before and independently by Renato Caccioppoli and Vladimir Dubrovskii: that theorem includes the ones of Nykodym, Vitali, Hahn and Saks and an earlier result of Gaetano Fichera [3], where a necessary and sufficient condition was proved for the integral with respect to a given fixed measure. The work of Cafiero is cited in the bibliography of the treatise on linear operators by Dunford and Schwartz but, to my knowledge, the only English reference discussing (very briefly) his contribution is the recent treatise of Vladimir Bogachev.

[1] Cafiero, F. (1953), "Sul passaggio al limite sotto il segno d'integrale per successioni d'integrali di Stieltjes-Lebesgue negli spazi astratti, con masse variabili con gli integrandi" [On the passage to the limit under the sign of integral for sequences of Stieltjes–Lebesgue integrals in abstract spaces, with masses varying jointly with integrands], Rendiconti del Seminario Matematico della Università di Padova (in Italian), 22: 223–245, MR 0057951, Zbl 0052.05003.

[2] Cafiero, F. (1959), Misura e integrazione [Measure and integration], (in Italian), Monografie matematiche del Consiglio Nazionale delle Ricerche 5, Roma: Edizioni Cremonese, pp. VII+451, MR 0215954, Zbl 0171.01503.

[3] Fichera, G. (1943), "Intorno al passaggio al limite sotto il segno d'integrale" [On the passage to the limit under the sign of integral], Portugaliae Mathematica (in Italian), 4 (1): 1–20, MR 0009192, Zbl 0063.01364.

[4] Letta, G. (2013), Argomenti scelti di Teoria della Misura [Selected topics in Measure Theory], (in Italian) Quaderni dell'Unione Matematica Italiana 54, Bologna: Unione Matematica Italiana, pp. XI+183, ISBN 88-371-1880-5, Zbl 1326.28001.