I have tried to find some explanations online but I couldn't understand them because they talk about "measure-spaces" and I don't know yet what this is.
I am currently reading a book about analysis. It is called Analysis 1 from Otto Forster. In the book the $p-$ norm is first introduced for vectors in $\mathbb{C}^n$. And the claim is now that one can generalize this $p-$ norm and the relating inequalities (Hölder, Minkowski) for integrals with the help of a theorem. But I don't know how I can employ this theorem to show the inequality.
I am gonna first name my problem and then list all the definitions in the book and theorems which I can use including the theorem mentioned above.
I hope someone can tell me how I can solve the problem with the things I already know.
Problem
Let $f:[a,b]\rightarrow \mathbb{R}$ be an integrable function and $p\in\mathbb{R}\geq 1$. Define $\|f\|_p:=(\int_a^b|f(x)|^pdx)^{1/p}$.
Show for integrable functions $f,g:[a,b]:\rightarrow \mathbb{R}$
$\|f+g\|_p\leq \|f\|_p+\|g\|_p,\forall p\geq 1$
$\int_a^b|f(x)g(x)|dx\leq \|f\|_p\|g\|_q$ with $p,q>1$ and $\frac{1}{p}+\frac{1}{q}=1$
What do I know
The current definition we use for integrals i.e. for an integrable function is that:
Let $a<b\in\mathbb{R}$. An integral is defined on a stepfunction $\phi:[a,b]\rightarrow \mathbb{R}$ i.e $\phi\in \mathcal{T}[a,b]$. $\int_a^b\phi(x)dx:= \sum_{k=1}^nc_k(x_k-x_{k-1})$. (I suppose that the definition of a stepfunction is well known and thus the meaning of $x_k,x_{k-1},c_k$)
We now define the upper-integral and lower-integral for a bounded function $f:[a,b]\rightarrow \mathbb{R}$
$\int_{a}^{b}{}^*f(x)dx:=\inf\{\int_{a}^b\phi(x)dx:\phi\in\mathcal{T}[a,b],\phi\geq f\}$. Analogously the lower-integral $\int_{a}^{b}{}_{*}f(x)dx$. A function is intregrable if upper and lower integral is the same. The integral of $f,\int_{a}^bf(x)dx$ is then defined as the upperintegral. Furthermore a function $f:[a,b]\rightarrow \mathbb{R}$ is integrable (necessarily bounded) iff for every $\epsilon>0$ there are stepfunctions $\phi,\psi\in\mathcal{T}[a,b]$ such that $\phi\leq f\leq \psi$ and $\int_{a}^b\psi(x)dx-\int_{a}^b\phi(x)dx\leq\epsilon$
We also know that for integrable functions $f,g$ and $\lambda\in\mathbb{R}$
$\int_{a}^b(f+g)(x)dx=\int_{a}^bf(x)dx+\int_{a}^bg(x)dx$
$\int_{a}^b(\lambda f)(x)dx=\lambda\int_{a}^bf(x)dx$
$f\leq g \Rightarrow \int_{a}^bf(x)dx\leq\int_{a}^bg(x)dx$
$|\int_{a}^bf(x)dx|\leq\int_{a}^b|f(x)|dx$
for every $p\in [1,\infty[ ,|f|^p$ is integrable
$fg:[a,b]\rightarrow \mathbb{R}$ is integrable
Mean value theorem for definite integrals.
We then also defined what a "Riemann-sum" is. Given a function $f:[a,b]\rightarrow \mathbb{R}$ and a subdivision of $[a,b]$
$a=x_0<x_1<...<x_n=b$ and $\xi_k\in[x_{k-1},x_k]$. We define $Z:=((x_k)_{0\leq k\leq n},(\xi_k)_{1\leq k\leq n})$. $S(Z,f):=\sum_{k=1}^nf(\xi_k)(x_k-x_{k-1})$ is called the "Riemann-sum" for $f$ in terms of $Z$. The mesh size of $Z$ is defined as $\mu(Z):=\max_{1\leq k\leq n}(x_k-x_{k-1})$.
The key theorem here is that: Let $f:[a,b]\rightarrow\mathbb{R}$ be a Riemann-integrable function. For every $\epsilon>0$ there exists $\delta>0$ such that for every choice of $Z$ ($(x_k)$ and $(\xi_k)$) with a mesh size $\mu(Z)\leq \delta$
$|\int_{a}^bf(x)dx-S(Z,f)|\leq \epsilon$
The $p$ norm for vectors $x=(x_1,...,x_n)\in\mathbb{C}^n$ is defined as $\|x\|_p:=(\sum_{v=1}^n|x_v|^p)^{1/p}$
We have proved that
(Hölder)
If $p,q\in]1,\infty[$ with $\frac{1}{p}+\frac{1}{q}=1$ then $\sum_{v=1}^n|x_vy_v|\leq\|x\|_p\|y\|_q$
(Minkowsky)
If $p\in[1,\infty[$ then for all $x,y\in\mathbb{C}^n$
$||x+y||_p\leq ||x||_p + ||y||_p$
Attempt
My idea was to was to take Riemann sums of the functions from the left side of the inequality and from the right side from the inequality and then using the already known Hölder inequality somehow and then say that because the Riemann sums were arbitrary chosen the theorem is true. But I am stuck and I don't know how to convert this idea into a proof and explain it properly. I really don't know how to approach the problem but since the author says it I think there is a connection with Riemann sums.
First consider that if the integral exists it holds $$\int_a^b f(x) dx = \lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n f(\xi_k)$$ with $\xi_k \in \left(\frac{k-1}{n},\frac{k}{n}\right)$ because the right hand side $$\frac{1}{n}\sum_{k=1}^n f(\xi_k)$$is nothing else then a Riemann sum for the equidistant mesh with mesh size $\frac{1}{n}$. and for equidistant meshes it's the same if we say the mesh size tends to zero or n tends to infinity.
And with the equality above we get: $$\begin{align}\|f+g\|_p &= \left(\int_a^b |(f + g)(x)|^p dx\right)^\frac{1}{p} \\ &= \lim_{n\to\infty} \frac{1}{n}\left(\sum_{k=1}^n |(f + g)(\xi_k)|^p\right)^\frac{1}{p} \\ &= \lim_{n\to\infty} \frac{1}{n}\left(\sum_{k=1}^n |f(\xi_k) + g(\xi_k)|^p\right)^\frac{1}{p} \\ \text{(Minkowski)}&\le \lim_{n\to\infty}\frac{1}{n}\left(\left(\sum_{k=1}^n |f(\xi_k)|^p\right)^\frac{1}{p} + \left(\frac{1}{n}\sum_{k=1}^n |g(\xi_k)|^p\right)^\frac{1}{p}\right) \\ &= \lim_{n\to\infty}\frac{1}{n}\left(\sum_{k=1}^n |f(\xi_k)|^p\right)^\frac{1}{p} + \lim_{n\to\infty}\frac{1}{n}\left(\sum_{k=1}^n |g(\xi_k)|^p\right)^\frac{1}{p} \\ &= \left(\int_a^b |f(x)|^p dx\right)^\frac{1}{p} + \left(\int_a^b |g(x)|^p dx\right)^\frac{1}{p} \\&= \|f\|_p+\|g\|_p\end{align}$$
Using the same argument we will get the Hölder inequality.