Die Funktion G besitze eine beschr?nkte Ableitung G^' auf [a, b] Dann ist G^' L-integrierbar und

step 1 Well, the integral is the opposite of the derivative. So if we start with F of X, G of X, G of X

Die Funktion G besitze eine beschr?nkte Ableitung G^' auf...

Die Funktion $G$ besitze eine beschr?nkte Ableitung $G^{\prime}$ auf $[a, b]$ Dann ist $G^{\prime}$ L-integrierbar und $$ G(b)=G(a)+\int_{a}^{b} G^{\prime} \mathrm{d} x $$ Anleitung zum Beweis: a) Setze $G(x):=G(b)$ für $x>b$ und $g_{n}(x):=\frac{G\left(x+\frac{1}{n}\right)-G(x)}{1 / n}$ für $x \in[a, b]$. b) Fïr alle $x \in[a, b]$ und alle $n \in \mathbf{N}$ ist $\left|g_{n}(x)\right| \leqslant \sup _{a<i<b}|G(t)|$ (das ist trivial für $x=b$ und folgt fûr $x \in[a, b)$ mit Hilfe des Mittelwettsatzes der Differentialrechnung). c) $\int_{a}^{b} g_{n} \mathrm{~d} x \rightarrow \int_{a}^{b} G^{\prime} \mathrm{d} x$ d) $\int_{a}^{b} G(x+1 / n) \mathrm{d} x=\int_{a+1 / n}^{b+1 / n} G(x) \mathrm{d} x$, also $$ \int_{a}^{b} g_{n}(x) \mathrm{d} x=n \int_{b}^{b+1 / n} G(x) \mathrm{d} x-n \int_{a}^{a+1 / n} G(x) \mathrm{d} x \rightarrow G(b)-G(a) $$ Beachte, $\mathrm{da} \beta G$ stetig ist!

Die Funktion $G$ besitze eine beschr?nkte Ableitung $G^{\prime}$ auf $[a, b]$ Dann ist $G^{\prime}$ L-integrierbar und
$$
G(b)=G(a)+\int_{a}^{b} G^{\prime} \mathrm{d} x
$$
Anleitung zum Beweis: a) Setze $G(x):=G(b)$ für $x>b$ und $g_{n}(x):=\frac{G\left(x+\frac{1}{n}\right)-G(x)}{1 / n}$ für $x \in[a, b]$.
b) Fïr alle $x \in[a, b]$ und alle $n \in \mathbf{N}$ ist $\left|g_{n}(x)\right| \leqslant \sup _{a<i<b}|G(t)|$ (das ist trivial für $x=b$ und folgt fûr $x \in[a, b)$ mit Hilfe des Mittelwettsatzes der Differentialrechnung).
c) $\int_{a}^{b} g_{n} \mathrm{~d} x \rightarrow \int_{a}^{b} G^{\prime} \mathrm{d} x$
d) $\int_{a}^{b} G(x+1 / n) \mathrm{d} x=\int_{a+1 / n}^{b+1 / n} G(x) \mathrm{d} x$, also
$$
\int_{a}^{b} g_{n}(x) \mathrm{d} x=n \int_{b}^{b+1 / n} G(x) \mathrm{d} x-n \int_{a}^{a+1 / n} G(x) \mathrm{d} x \rightarrow G(b)-G(a)
$$
Beachte, $\mathrm{da} \beta G$ stetig ist!

Key Concepts

Bounded Derivative

A function with a bounded derivative on an interval is controlled in its rate of change, which is central to many aspects of real analysis. In this context, the boundedness ensures that the difference quotients used to approximate the derivative are uniformly bounded, facilitating further analysis such as integrability and convergence arguments.

Fundamental Theorem of Calculus

This theorem establishes the direct connection between differentiation and integration. It states that if a function is differentiable and its derivative is integrable on an interval, then the integral of the derivative over that interval recovers the change in the function’s values. This principle is key in reasoning that G(b) equals G(a) plus the integral of G' from a to b.

Mean Value Theorem

The Mean Value Theorem is an essential tool in proving properties of functions with derivatives. It guarantees that for any two points there exists an intermediate point where the derivative equals the average rate of change of the function. This underlies the construction and estimation of difference quotients in the proof.

Dominated Convergence Theorem

Within the framework of Lebesgue integration, the Dominated Convergence Theorem allows one to interchange limits and integrals when the sequence of functions converges pointwise and is uniformly bounded by an integrable function. This concept is pivotal in justifying the convergence of the integrals of the approximating difference quotients to the integral of the derivative.

Translation Invariance in Integration

This concept refers to the property that integrals of a function over an interval can be shifted to a translated interval without affecting the integral's value, provided the function behaves appropriately. In the proof, a change of variables via translation helps to relate the integrals of shifted versions of the function back to the original interval, thereby connecting the approximations to the desired result.

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