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 $X$.

Comments

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 Template:ZBL
[a2]	W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948) ((Appendix by L.S. Pontryagin and L.G. Shnirel'man in Russian edition.)) MR Template:ZBL
[a3]	C. Kuratowski, "Introduction to set theory and topology" , Pergamon (1972) (Translated from Polish) MR0346724 Template:ZBL Template:ZBL

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 Template:ZBL Template:ZBL
[2]	H. Lebesgue, "Intégrale, longueur, aire" , Univ. Paris (1902) (Thesis) MR Template:ZBL
[3]	B. Levi, "Sopra l'integrazione delle serie" Rend. Ist. Lombardo sue Lett. (2) , 39 (1906) pp. 775–780 MR Template:ZBL
[4]	S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French) MR0167578 Template:ZBL Template:ZBL Template:ZBL
[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 Template:ZBL

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

[a1]	N. Dunford, J.T. Schwartz, "Linear operators" , 1–3 , Interscience (1958–1971) MR1009164 MR1009163 MR1009162 MR0412888 MR0216304 MR0188745 MR0216303 MR1530651 MR0117523 Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL Template:ZBL
[a2]	P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Template:ZBL
[a3]	E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Template:ZBL