I am trying to show the equivalence of two definitions of the Lebesgue-integral for functions $f\colon\mathbb{R}^k\to\mathbb{R}$.
The first book (Königsberger, Analysis 2, Springer) defines a semi-norm for a function $f\colon \mathbb{R}^k\to\mathbb{R}\cup\{∞\}$ by setting $\Vert f\Vert_1 = \inf\{I(\phi) : |f(x)|≤\phi(x)\}∈\mathbb{R}_0^+\cup\{\infty\}$ where $\phi$ is a infinite sum
$$ \phi = \sum_{k=1}^∞ c_k \mathbb{1}_{Q_k}, $$
the $Q_i$ are open cubes in $\mathbb{R}^n$, $c_i≥ 0$ and $I(\phi)$ is the canonic extension of the definition of an integral for a step function. Now we say that $f$ is Lebesgue integrable if there exists a sequence of step functions $(\phi_n)_n$ such that $\Vert f-\phi_n\Vert_1 \to 0$ when $n\to∞$. In this case you define $$∫ f := \lim ∫\phi_n.$$
The other book (A. Weir, Lebesgue Integration and Measure, CUP) also defines an integral $I(\phi)$ for step functions and says that a function $f\colon\mathbb{R}^k\to\mathbb{R}$ belongs to $L^+$ if there exists an increasing sequence $(\phi)_n$ of step functions converging to $f$ almost everywhere (so null sets are already defined without a measure by covering the set with bounded intervals so that the total length of these intervals is as small as you like) so that the sequence $(∫\phi_n)_n$ of integrals converges and then you set $$∫ f := \lim \int \phi_n.$$ Then a function $h\colon\mathbb{R}^k\to\mathbb{R}$ is said to be Lebesgue integrable if it is the difference of two functions $a,b∈L^+$, i.e. $h=a-b$, and then you have the definition $∫ h= ∫ a-∫ b$.
My question is if you know how to show the equivalence of these two approaches to the Lebesgue integral or if you know some literture where I can find a proof.
Thank you very much for your help!
well ... you need to establish that any function that is Lebesgue integrable according to one definition is also Lebesgue integrable according to the other definition. Here's a high-level sketch of the proof:
1. From Königsberger's Definition to Weir's Definition:
Let $ f: \mathbb{R}^k \rightarrow \mathbb{R} \cup \{\infty\} $ be Lebesgue integrable according to Königsberger's definition. This means there exists a sequence of step functions $ \{\phi_n\} $ such that $ \|f - \phi_n\|_1 \rightarrow 0 $ as $ n \rightarrow \infty $.
Define $ a_n = \inf\{I(\psi) : |f(x)| \leq \psi, \psi \text{ is a step function}\} $, and let $ \{a_n\} $ be the corresponding sequence of integrals.
Construct an increasing sequence of step functions $ \{\psi_n\} $ such that $a_n \leq I(\psi_n) \leq a_{n+1} $ for all $ n $. This can be achieved by considering simple step functions that approximate $ f $ from below.
Now, according to Weir's definition, $ f $ is Lebesgue integrable, and $ \int f = \lim \int \psi_n $.
2. From Weir's Definition to Königsberger's Definition:
Assume $ f $ is Lebesgue integrable according to Weir's definition, i.e., there exist $ a, b \in L^+ $ such that $ f = a - b $. Let $ \{\phi_n\} $ be an increasing sequence of step functions converging to $ a $ almost everywhere. Similarly, let $ \{\psi_n\} $ be an increasing sequence of step functions converging to $ b $ almost everywhere.
Construct a sequence of step functions $ \{\chi_n\} $ by setting $ \chi_n = \phi_n + \psi_n $. Note that $ |\chi_n| \leq \phi_n + \psi_n \leq a + b $ for all $ n $.
According to Königsberger's definition, $ \|f - \chi_n\|_1 \rightarrow 0 $ as $ n \rightarrow \infty $.