I have proved this version by applying Hölder's inequality . Could you confirm if my proof is correct? I also would like to ask for a non-measure-theoretic proof.
For $p,q \ge 0$ such that $1/p+1/q=1$ and $a_1, \ldots, a_n,b_1, \ldots, b_n \ge 0$, $$\sum_{k=1}^{n} a_kb_k \leq \left (\sum_{k=1}^{n} a_k^{p} \right)^{1/p} \left (\sum\limits_{k=1}^{n} b_k^{q} \right)^{1/q}.$$
My proof: We apply the following Hölder's inequality
Let
$(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, \| \cdot \|)$ a Banach space.
$\|\cdot \|_p$ the $L_p$-norm of $L_p(X, \mu,E)$.
$f,g: X \to E$ are $\mu$-measurable.
$p,q >0$ such that $1/p+1/q = 1$.
Then $$\int_{X} {\| f (x)\|} \cdot {\|g (x)\|}{} \mathrm d \mu (x) \leq \|f\|_p \cdot \|g\|_q.$$
Let $[n] \triangleq \{1, \ldots, n\}$ and $([n], \mathcal P([n]), \mu)$ be the finite counting measure space. We define $f,g: [n] \to \mathbb R_+$ by $f(k) \triangleq a_k$ and $g(k) \triangleq b_k$. Clearly, $f$ and $g$ are measurable and non-negative. We have \begin{align} \int_{[n]} {\| f (k)\|} \cdot {\|g (k)\|} \mathrm d \mu (k) &= \sum_{k=1}^{n} a_kb_k \\ \|f\|_p \cdot \|g\|_q &= \left ( \int_{[n]} f^p(k) \mathrm d \mu (k) \right )^{1/p} \left ( \int_{[n]} g^q(k) \mathrm d \mu (k) \right )^{1/q} \\ &= \left (\sum_{k=1}^{n} a_k^{p} \right)^{1/p} \left (\sum\limits_{k=1}^{n} b_k^{q} \right)^{1/q}. \end{align}
This completes the proof.