View source for Lebesgue theorem

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Lebesgue's theorem in dimension theory: For any $\epsilon>0$ the $n$-dimensional cube has a finite closed $\epsilon$-covering of multiplicity $\leq n+1$, and at the same there is an $\epsilon_0=\epsilon_0(n)>0$ such that any finite closed $\epsilon_0$-covering of the $n$-dimensional cube has multiplicity $\geq n+1$ (cf. also [[Covering (of a set)|Covering (of a set)]]). This assertion led later to a definition of a fundamental dimension invariant, the [[Lebesgue dimension|Lebesgue dimension]] $\dim X$ of a normal topological space $X$.



====Comments====
This theorem is also called the Lebesgue covering theorem or  "Pflastersatz"  (see [[Dimension|Dimension]]). In the language of [[Dimension theory|dimension theory]] it says that $\dim I^n=n$ for every $n$.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,   "Dimension theory" , North-Holland &amp; PWN  (1978)  pp. 19; 50  {{MR|0482696}} {{MR|0482697}} {{ZBL|0401.54029}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Hurevicz,   G. Wallman,   "Dimension theory" , Princeton Univ. Press  (1948)  ((Appendix by L.S. Pontryagin and L.G. Shnirel'man in Russian edition.))  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C. Kuratowski,   "Introduction to set theory and topology" , Pergamon  (1972)  (Translated from Polish)  {{MR|0346724}} {{ZBL|0267.54002}} {{ZBL|0247.54001}} </TD></TR></table>

Lebesgue's theorem on the passage to the limit under the integral sign: Suppose that on a measurable set $E$ there is specified a sequence of measurable functions $f_n$ that converges almost-everywhere (or in measure) on $E$ to a function $f$. If there is a summable function $\Phi$ on $E$ such that for all $n$ and $x$,

$$|f_n(x)|\leq\Phi(x),$$

then $f_n$ and $f$ are summable on $E$ and

$$\lim_{n\to\infty}\int\limits_Ef_n(x)\,dx=\int\limits_Ef(x)\,dx.$$

This was first proved by H. Lebesgue [[#References|[1]]]. The important special case when $\Phi=\text{const}$ and $E$ has finite measure is also called the Lebesgue theorem; he obtained it earlier [[#References|[2]]].

A theorem first proved by B. Levi [[#References|[3]]] is sometimes called the Lebesgue theorem: Suppose that on a measurable set $E$ there is specified a non-decreasing sequence of measurable non-negative functions $0\leq f_1(x)\leq f_2(x)\leq\dots$ ($x\in E$) and that

$$f(x)=\lim_{n\to\infty}f_n(x)$$

almost-everywhere; then

$$\lim_{n\to\infty}\int\limits_Ef_n(x)\,dx=\int\limits_Ef(x)\,dx.$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Lebesgue,   "Sur les intégrales singuliéres"  ''Ann. Fac. Sci. Univ. Toulouse Sci. Math. Sci. Phys.'' , '''1'''  (1909)  pp. 25–117  {{MR|1508308}}  {{ZBL|41.0329.01}}  {{ZBL|41.0327.02}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Lebesgue,   "Intégrale, longueur, aire" , Univ. Paris  (1902)  (Thesis)  {{MR|}}  {{ZBL|33.0307.02}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B. Levi,   "Sopra l'integrazione delle serie"  ''Rend. Ist. Lombardo sue Lett. (2)'' , '''39'''  (1906)  pp. 775–780  {{MR|}}  {{ZBL|37.0424.03}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Saks,   "Theory of the integral" , Hafner  (1952)  (Translated from French)  {{MR|0167578}} {{ZBL|1196.28001}} {{ZBL|0017.30004}}  {{ZBL|63.0183.05}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.P. Natanson,   "Theory of functions of a real variable" , '''1–2''' , F. Ungar  (1955–1961)  (Translated from Russian)  {{MR|0640867}} {{MR|0354979}} {{MR|0148805}} {{MR|0067952}} {{MR|0039790}} {{ZBL|}} </TD></TR></table>

''T.P. Lukashenko''

====Comments====
This Lebesgue theorem is also called the dominated convergence theorem, while Levi's theorem is also known as the monotone convergence theorem.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators" , '''1–3''' , Interscience  (1958–1971)  {{MR|1009164}} {{MR|1009163}} {{MR|1009162}} {{MR|0412888}} {{MR|0216304}} {{MR|0188745}} {{MR|0216303}} {{MR|1530651}} {{MR|0117523}} {{ZBL|0635.47003}} {{ZBL|0635.47002}} {{ZBL|0635.47001}} {{ZBL|0283.47002}} {{ZBL|0243.47001}} {{ZBL|0146.12601}} {{ZBL|0128.34803}} {{ZBL|0084.10402}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Halmos,   "Measure theory" , v. Nostrand  (1950)  {{MR|0033869}} {{ZBL|0040.16802}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Hewitt,   K.R. Stromberg,   "Real and abstract analysis" , Springer  (1965)  {{MR|0188387}} {{ZBL|0137.03202}} </TD></TR></table>