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MATHEMATICS
OSWALD DEMUTH
ON LEBESGUE INTEGRATION IN CONSTRUCTIVE ANALYSIS
(Presented by Academician P. S. Novikov on 11 XI 1964)
This note is devoted to problems of Lebesgue integration on the segment \(0 \triangle 1\) of constructive functions of a real variable (c.f.r.v.), defined everywhere on \(0 \triangle 1\). It uses the definitions and results of the papers \((^{1-9})\).
In what follows the letters \(a, b\) serve as variables for rational numbers, \(x\) as a variable for real duplexes \((^4)\).
An exact disjunctive segment covering \(\Phi\) of the segment \(0 \triangle 1\) \((^8)\) will be called simply a covering, if \(\vartheta_{\mathrm{l}}(\Phi_0)=0\) and \(\vartheta_{\mathrm{r}}(\Phi_1)=1\). Everywhere, a defined c.f.r.v. \(f\) will be called simply a function, if
\[ \forall x\,((x \leqslant 0 \supset f(x)=f(0))\ \&\ (x \geqslant 1 \supset f(x)=f(1))). \]
It is clear that, in studying integration problems for the indicated class of c.f.r.v., it is sufficient to restrict oneself to considering functions.
For a function \(f\) and a covering \(\Phi\), denote by \(f/\Phi\) the function such that
\[ \forall kx\left(x\in\Phi_k \supset (f/\Phi)(x)= f(\vartheta_{\mathrm{l}}(\Phi_k))+ \frac{f(\vartheta_{\mathrm{r}}(\Phi_k))-f(\vartheta_{\mathrm{l}}(\Phi_k))}{|\Phi_k|} (x-\vartheta_{\mathrm{l}}(\Phi_k))\right). \]
By pseudopolygonal functions we shall mean functions which are pseudopolygonal in the sense of \((^5)\). By systems of constructive objects of the given type we shall mean systems of words \((^1)\) of this type, and by sequences of functions (coverings) normal algorithms \(\mathfrak{A}\) such that, for every natural \(n\), \(\mathfrak{A}_n\) is a function (covering).
Definition 1. Let \(\alpha\) be a letter not belonging to the alphabet \(\{0,\ |,\ -,\ /,\ \diamond\}\). Words of the form
\[ \alpha a_0 \alpha a_1 \alpha \ldots \alpha a_n \alpha b_0 \alpha b_1 \alpha \ldots \alpha b_n \alpha, \]
where \(n\) is a natural number, \(a_i\) and \(b_i\) are given rational numbers \((0\leqslant i\leqslant n)\), \(0=a_0<a_1<\cdots<a_n=1\), will be called polygonal bases.
One can construct normal algorithms \(\mathfrak{K}\) and \(\mathfrak{I}\) such that, for every polygonal basis \(F\), where
\[ F=\alpha a_0\alpha a_1\alpha\ldots\alpha a_n\alpha b_0\alpha b_1\alpha\ldots\alpha b_n\alpha, \]
\(\mathfrak{K}_F\) is a function and
\[ \forall ix\left(1\leqslant i\leqslant n\ \&\ a_{i-1}\leqslant x\leqslant a_i \supset \mathfrak{K}_F(x)=a_{i-1}+ \frac{b_i-b_{i-1}}{a_i-a_{i-1}}(x-a_{i-1})\right); \]
\(\mathfrak{I}_F\) is an everywhere defined and everywhere differentiable c.f.r.v. for which
\[ \forall x\,((\mathfrak{I}_F)'(x)=\mathfrak{K}_F(x)). \]
For polygonal skeletons one can define the operations of addition (denote it by \(\dot{+}\)), subtraction (\(\dot{-}\)), and absolute value (\(|\ |_{\circ}\)) so that, for any polygonal skeletons \(F\) and \(G\) and any real duplex \(x\), one has
\[ \widetilde{\mathfrak{R}}_{(F\dot{+}G)}(x)=\widetilde{\mathfrak{R}}_{F}(x)+\widetilde{\mathfrak{R}}_{G}(x),\qquad \widetilde{\mathfrak{R}}_{(F\dot{-}G)}(x)=\widetilde{\mathfrak{R}}_{F}(x)-\widetilde{\mathfrak{R}}_{G}(x), \]
\[ \widetilde{\mathfrak{R}}_{|F|_{\circ}}(x)=\left|\widetilde{\mathfrak{R}}_{F}(x)\right|. \]
Definition 2. Let \(f\) be a function. We shall say that \(f\) is Lebesgue integrable on the segment \(0\triangle 1\) and write \(f\in(\mathcal{L})\), if there exists a sequence of polygonal skeletons \(\{F_n\}_n\) such that
\[ \forall xm\,\exists k\,\forall n\left(n\geq k\supset \left|\widetilde{\mathfrak{R}}_{F_n}(x)-f(x)\right|<\frac{1}{m+1}\right), \tag{1} \]
\[ \forall m\,\exists l\,\forall nk\left(n\geq l\supset \widetilde{\mathfrak{J}}_{F_n\dot{-}F_{n+k}|_{\circ}}(1)- \widetilde{\mathfrak{J}}_{F_n\dot{-}F_{n+k}|_{\circ}}(0)<\frac{1}{m+1}\right). \]
It is clear that for functions \(f,g\) such that \(f\in(\mathcal{L})\), \(g\in(\mathcal{L})\), and for a real duplex \(y\), one has \(|f|\in(\mathcal{L})\), \((f+g)\in(\mathcal{L})\), \((y\cdot f)\in(\mathcal{L})\).
Theorem 1. Let \(f\) be a function of weakly bounded variation on \(0\triangle 1\) \((^8)\). Then \(f\) is Riemann integrable on \(0\triangle 1\) (for the definition see \((^9)\)), \(f\in(\mathcal{L})\).
Definition 3. For a function \(f\), \(f\in(\mathcal{L})\), and real duplexes \(v,y,z\), where \(0\leq y\leq 1\) and \(0\leq z\leq 1\), we shall say that \(v\) is the value of the Lebesgue integral of \(f\) with limits \(y,z\), and write
\[ v=\int_y^z f\,dx, \]
if there exists a c.f.d.s. \(\mathfrak{F}\), defined everywhere on \(0\triangle 1\) and differentiable on \(0\nabla 1\), for which \(\forall x(0<x<1\supset \mathfrak{F}'(x)=f(x))\), \(v=\mathfrak{F}(z)-\mathfrak{F}(y)\).
We shall also use the customary abbreviated notation, which we explain by an example. Let \(f\) be a function, \(f\in(\mathcal{L})\); let \(y,z\) be real duplexes, \(0\leq y\leq 1\), \(0\leq z\leq 1\); let \(P(v)\) be a predicate. Suppose that no occurrence of \(v\) in \(P\) lies in the scope of a quantifier binding \(v,y,z\). Then we write
\[ P\left(\int_y^z f\,dx\right)\Longleftrightarrow \forall v\left(v=\int_y^z f\,dx\supset P(v)\right). \]
Theorem 2. Let \(f\) be a function, \(f\in(\mathcal{L})\). Then there exists an everywhere-defined and everywhere differentiable c.f.d.s. \(\mathfrak{F}\) such that \(\forall x(\mathfrak{F}'(x)=f(x))\), \(\mathfrak{F}\) is a c.f.d.s. of bounded variation on \(0\triangle 1\), and
\[ \bigvee_{0}^{1}\mathfrak{F}=\int_0^1 |f|\,dx. \]
If \(\{F_n\}_n\) is a sequence of polygonal skeletons such that (1) is satisfied, then
\[ \forall m\,\exists k\,\forall nx\left(n\geq k\supset \left|\widetilde{\mathfrak{J}}_{F_n}(x)-\mathfrak{F}(x)\right|\leq \frac{1}{m+1}(|x|+1)\right). \]
Theorem 3. Let \(f\) be a function. Then \(f\in(\mathcal{L})\) if and only if there exist a c.f.d.s. \(\mathfrak{F}\), defined everywhere on \(0\triangle 1\) and differentiable on \(0\nabla 1\), and a sequence of polygonal skeletons \(\{T_n\}_n\) such that \(\forall x(0<x<1\supset \mathfrak{F}'(x)=f(x))\), and for every system of rational numbers \(\{c_i\}_{i=0}^{p}\), for which \(0\leq c_0\leq c_1\leq\cdots\leq c_p\leq 1\), and every natural \(n\),
\[ \sum_{i=1}^{p}\left| \left(\mathfrak{F}(c_i)-\widetilde{\mathfrak{R}}_{T_n}(c_i)\right)- \left(\mathfrak{F}(c_{i-1})-\widetilde{\mathfrak{R}}_{T_n}(c_{i-1})\right) \right|<\frac{1}{n+1}. \]
Theorem 4. Let $\{f_n\}_n$ be a sequence of functions, and let $f$ be a function, such that
$\forall n\,(f_n\in(\mathcal L))$,
$\forall x m\,\exists k\,\forall n\left(n\geq k\supset |f_n(x)-f(x)|<\dfrac{1}{m+1}\right)$, and
\[ \forall m\,\exists l\,\forall nk\left(n\geq l\supset \int_0^1 |f_n-f_{n+k}|\,dx<\frac{1}{m+1}\right). \]
Then $f\in(\mathcal L)$ and
\[ \forall m\,\exists l\,\forall n\left(n\geq l\supset \int_0^1 |f_n-f|\,dx<\frac{1}{m+1}\right). \]
Theorem 5. Let $f$ be a function, $f\in(\mathcal L)$, and let $n$ be a natural number. Then there exists a pseudopolygonal function $g$ such that $g\in(\mathcal L)$ and
\[ \forall x\left(|g(x)-f(x)|<\frac{1}{n+1}\right). \]
Theorem 6. Let $f$ be a pseudopolygonal function. Then $f\in(\mathcal L)$ if and only if there exists a sequence of coverings $\{\,{}^n\Phi\,\}_n$ such that, for every natural $n$, $f/{}^n\Phi$ is a uniformly continuous function,
\[ \forall k\left(\sum_{i=0}^{k} |\,{}^n\Phi_i\,|<\frac{1}{n+1}\right), \]
$\forall k\,\exists l\left({}^{\,n+1}\Phi_k\subseteq {}^n\Phi_l\right)$, and
\[ \forall m\,\exists l\,\forall nk\left(n\geq l\supset \int_0^1 |\,f/{}^n\Phi-f/{}^{n+k}\Phi\,|\,dx<\frac{1}{m+1}\right). \]
Theorem 7. Let $f,g$ be functions, $f\in(\mathcal L)$,
$\forall ab\left(|g(a)-g(b)|\leq |f(a)-f(b)|\right)$. Then $g\in(\mathcal L)$.
Theorem 8. Let $f,g$ be functions and let $\{g_n\}_n$ be a sequence of functions such that $f\in(\mathcal L)$,
\[ \forall nab\left(|g_n(a)-g_n(b)|\leq |f(a)-f(b)|\right), \]
\[ \forall xm\,\exists k\,\forall n\left(n\geq k\supset |g_n(x)-g(x)|<\frac{1}{m+1}\right). \]
Then $g\in(\mathcal L)$, $\forall n\,(g_n\in(\mathcal L))$, and
\[ \forall m\,\exists k\,\forall n\left(n\geq k\supset \int_0^1 |g_n-g|\,dx<\frac{1}{m+1}\right). \]
Moscow State University
named after M. V. Lomonosov
Received
11 XI 1964
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