# Lebesgue theorem

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)). This assertion led later to a definition of a fundamental dimension invariant, the Lebesgue dimension $\dim X$ of a normal topological space $X$.

## Contents

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

#### References

 [a1] R. Engelking, "Dimension theory" , North-Holland & PWN (1978) pp. 19; 50 MR0482696 MR0482697 Zbl 0401.54029 [a2] W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948) ((Appendix by L.S. Pontryagin and L.G. Shnirel'man in Russian edition.)) [a3] C. Kuratowski, "Introduction to set theory and topology" , Pergamon (1972) (Translated from Polish) MR0346724 Zbl 0267.54002 Zbl 0247.54001

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 [1]. The important special case when $\Phi=\text{const}$ and $E$ has finite measure is also called the Lebesgue theorem; he obtained it earlier [2].

A theorem first proved by B. Levi [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

 [1] H. Lebesgue, "Sur les intégrales singuliéres" Ann. Fac. Sci. Univ. Toulouse Sci. Math. Sci. Phys. , 1 (1909) pp. 25–117 MR1508308 Zbl 41.0329.01 Zbl 41.0327.02 [2] H. Lebesgue, "Intégrale, longueur, aire" , Univ. Paris (1902) (Thesis) Zbl 33.0307.02 [3] B. Levi, "Sopra l'integrazione delle serie" Rend. Ist. Lombardo sue Lett. (2) , 39 (1906) pp. 775–780 Zbl 37.0424.03 [4] S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) MR0167578 Zbl 1196.28001 Zbl 0017.30004 Zbl 63.0183.05 [5] I.P. Natanson, "Theory of functions of a real variable" , 1–2 , F. Ungar (1955–1961) (Translated from Russian) MR0640867 MR0354979 MR0148805 MR0067952 MR0039790

T.P. Lukashenko