(Equivalence Classes Family $L^p$). Let $(X,\mathcal{A},\mu )$ be a measure space [7]. Let $p\in \mathbb{R}$ such that $1\le p<\infty$. Let ${\mathcal{L}}^p\left(\mu \right)$ be a family of measurable functions [7] $X\to \mathbb{R}$ such that Suppose an equivalence relation [4] ∼ on ${\mathcal{L}}^p\left(\mu \right)$ defined by Then we have a family of equivalence classes [4] in ${\mathcal{L}}^p\left(\mu \right)$ in which

(Vector Space). A set V is a vector space [3] over a field $\mathbb{F}$ together with operations $+:V\times V\to V$ and $·:\mathbb{F}\times V\to V$ if the following axioms are satisfied:

To show that $L^p\left(\mu \right)$ is a vector space over $\mathbb{R}$ we need to show that all axioms in definition 2 hold.

Let $f,g,h\in L^p\left(\mu \right)$. Since $f,g,h$ are real-valued functions, then and holds and show that axioms V1 and V2 are satisfied. Note that in the sense of $L^p\left(\mu \right)$, the symbol ‘=’ designates the equivalence relation ∼ as described in definition 1. The zero function $e\in L^p\left(\mu \right)$ also exists as it satisfies Then we have which shows that axiom V3 is satisfied. Then $-f\in L^p\left(\mu \right)$ is certain since With axiom V4 is satisfied. Note that since Now, let $a,b\in \mathbb{R}$. Then we obtain which shows that axiom V5 is satisfied. Then, certainly $1\in \mathbb{R}$ and we have which shows that axiom V6 is satisfied. Also and show that axioms V7 and V8 are satisfied. Thus, we conclude that $L^p\left(\mu \right)$ is a vector space. □

(Normed Space). Let V be a vector space over a field $\mathbb{F}$. A norm on V is a map $\parallel ·\parallel :V\to [0,\infty )$ satisfying the following axioms:

Let $a,b\in \mathbb{R}$ such that $a,b\ge 0$. Let $p,q>1$ such that $\frac{1}{p}+\frac{1}{q}=1$. Then the inequality known as Young’s inequality [2], holds.

Suppose a function $f:[0,a]\to [0,\alpha ]$ such that $b\le \alpha$, $f\left(a\right)=\alpha$ and f is continuous [8] as well as bijective [4]. Note that we will have In particular, f can be defined by as it is continuous [2] and bijective. And we can show the bijectivity of f if we can show that f is strictly monotone, since strict monotonicity of such a function implies bijectivity. Now let $s,t\in [0,a]$ such that $s<t$. Certainly we have which shows that f is strictly increasing. Hence f is bijective. It implies that $f^{-1}$ exists, which is now our task in finding it. Follows from proposition 1, we obtain the inverse $f^{-1}$ as given by And we have which proves the lemma. □

Let $(X,\mathcal{A},\mu )$ be a measure space [7]. Let $p,q>1$ such that $\frac{1}{p}+\frac{1}{q}=1$. Let $f\in L^p\left(\mu \right)$ and $g\in L^q\left(\mu \right)$ [7]. Then holds, which is known as Hölder’s inequality.

Suppose some functions $\tilde{f},\tilde{g}:X\to \mathbb{R}$ defined by and respectively. Follows from lemma 1, i. e., Young’s inequality, we obtain By taking Lebesgue integral [7] to the expression above, we obtain which implies which proves Hölder’s inequality. □

Let $(X,\mathcal{A},\mu )$ be a measure space [7]. Let $p\in \mathbb{R}$ such that $1\le p<\infty$. Follows from theorem 1, we have a vector space $L^p\left(\mu \right)$. Let $f,g\in L^p\left(\mu \right)$. Then holds, which is known as Minkowski inequality [2].

Proving for $p=1$ is trivial, since obviously then by the property of Lebesgue integral [7] we have which proves the inequality for $p=1$.

Now we prove the inequality for $p>1$. Let $q\in \mathbb{R}$ such that $q>1$ and $\frac{1}{p}+\frac{1}{q}=1$. Let $h:X\to \mathbb{R}$ such that Note that $|f+g|\in L^p\left(\mu \right)$ since $L^p\left(\mu \right)$ is a vector space, which means that it is closed under addition and scalar multiplication. That is, where the map $\mathrm{sgn}:\mathbb{R}\to \{-1,0,1\}$ is defined by From the expression above, additive closure shows that $f+g\in L^p\left(\mu \right)$, and scalar multiplicative closure shows that $\mathrm{sgn}(f+g)·(f+g)\in L^p\left(\mu \right)$, and hence $|f+g|\in L^p\left(\mu \right)$. Follows from proposition 1, we obtain from the fact that $|f+g|\in L^p\left(\mu \right)$. Hence $h\in L^q\left(\mu \right)$. Then follows from the triangle inequality and Hölder’s inequality we obtain which implies which proves Minkowski inequality for $p>1$. Thus, we conclude the proof. □

Let $(X,\mathcal{A},\mu )$ be a measure space [7]. Let $p\in \mathbb{R}$ such that $1\le p<\infty$. Follows from theorem 1, we have a vector space $L^p\left(\mu \right)$. The expression holds.

We prove the forward part first. Let $a\in \mathbb{R}$ and $f\in L^p\left(\mu \right)$. Suppose $af\sim 0$. Then we have which implies And by the property of Lebesgue integral [7], we obtain which proves the forward part.

Now for the backward part, suppose Let $g:X\to \mathbb{R}$ such that $g:={\left|af\right|}^p$. Note that g is measurable and nonnegative. By simple function [7] approximation theorem, for every $n\in \mathbb{N}$ there exists a simple function $s_n:X\to [0,\infty )$ such that and holds. Note that there are only finitely many points in the image of $s_n$ since it is a simple function, for any $n\in \mathbb{N}$. Now let such that The notation $\mathrm{im}$ above refers to the image of a function [4]. Now suppose And suppose We can now express the simple function as where which is known as an indicator function. And the Lebesgue integral of g over X can be given in accordance with its definition [7] by Recalling the assumption, we now have Note that each term in the summation above is nonnegative. Then we must have which implies The only solution to this circumstance is The statement above implies that which means that $g\sim 0$, or $g=0$ μ-almost everywhere [7]. It also implies that $af\sim 0$, which proves the backward part. We conclude that which proves the theorem as a whole. □

Let $(X,\mathcal{A},\mu )$ be a measure space [7]. Let $p\in \mathbb{R}$ such that $1\le p<\infty$. Follows from theorem 1, we have a vector space $L^p\left(\mu \right)$. A map ${\parallel ·\parallel }_p:L^p\left(\mu \right)\to [0,\infty )$ defined by is a norm on $L^p\left(\mu \right)$ known as the p-norm, and hence $(L^p\left(\mu \right),\parallel ·{\parallel }_p)$ is a normed space.

To show that ${\parallel ·\parallel }_p$ is a norm on $L^p\left(\mu \right)$, we need to show that axioms N1, N2, N3 and N4 in definition 3 are satisfied. For the whole discussion in this proof, let $a\in \mathbb{R}$ and $f,g\in L^p\left(\mu \right)$.

By definition, ${\parallel ·\parallel }_p$ involves the designation of ${|·|}^p$ within the Lebesgue integral, hence we will always have a nonnegative integrand. By the property of Lebesgue integral [7], since ${\left|f\right|}^p\ge 0$ then which shows that axiom N1 is satisfied.

Now we need to show that ${\parallel f\parallel }_p=0\iff f=0$, which is equivalent to showing ${\parallel f\parallel }_p^p=0\iff f=0$, as shown by Follows from lemma 2, we obtain Note that the equality symbol ‘=’ in terms of $L^p\left(\mu \right)$ refers to ’∼’ in terms of ${\mathcal{L}}^p\left(\mu \right)$. Thus, the statement above shows that ${\parallel f\parallel }_p^p=0\iff f=0$ which implies ${\parallel f\parallel }_p=0\iff f=0$ and hence axiom N2 is satisfied.

By the property of Lebesgue integral [7], we obtain which shows that axiom N3 is satisfied.

Then by the property of Lebesgue integral as well as theorem 3 (Minkowski inequality), we obtain which shows that axiom N4 is satisfied.

Thus, we conclude that ${\parallel ·\parallel }_p:L^p\left(\mu \right)\to \mathbb{R}$ is a norm on $L^p\left(\mu \right)$. Hence $(L^p\left(\mu \right),\parallel ·{\parallel }_p)$ is a normed space. □

Let $(X,\mathcal{A},\mu )$ be a measure space [7]. Let $p\in \mathbb{R}$ such that $1\le p<\infty$. Follows from theorem 1, we have a vector space $L^p\left(\mu \right)$. Let $r\in L^p\left(\mu \right)$ such that i. e., $r\ne 0$ μ-almost everywhere [7]. Let A map ${\parallel ·\parallel }_{p:r}:L^p\left(\mu \right)\to [0,\infty )$ defined by is a norm on $L^p\left(\mu \right)$.

We need to show that axioms N1, N2, N3 and N4 are satisfied. For the whole disucssion in this proof, let $a\in \mathbb{R}$ and $f,g\in L^p\left(\mu \right)$.

Note that $r\in L^p\left(\mu \right)$, $r\ne 0$ μ-almost everywhere [7] and ${\left|r\right|}^p\ge 0$, which imply ${0<\left|r\right|}^p<\infty$ μ-almost everywhere. Then And note that $f\in L^p\left(\mu \right)$, which means Then by the property of Lebesgue integral [7], we obtain which shows that axiom N1 is satisfied.

Note that and hold. Follows from lemma 2, we obtain Note that the equality symbol ’=’ in terms of $L^p\left(\mu \right)$ refers to the equivalence relation [4] symbol ’∼’ in terms of ${\mathcal{L}}^p\left(\mu \right)$. Then by the transitive property of biconditional statements, we obtain which shows that axiom N2 is satisfied.

By the property of Lebesgue integral [7], we obtain which shows that axiom N3 is satisfied.

Follows from the property of Lebesgue integral [7] and theoerm 3 (Minkowski inequality), we obtain which shows that axiom N4 is satisfied. We conclude that the map ${\parallel ·\parallel }_{p:r}:L^p\left(\mu \right)\to [0,\infty )$ is a norm on $L^p\left(\mu \right)$. □

(r-Normed Space). Let $(X,\mathcal{A},\mu )$ be a measure space [7]. Let $p\in \mathbb{R}$ such that $1\le p<\infty$. Follows from theorem 1, we have a vector space $L^p\left(\mu \right)$. Let $r\in L^p\left(\mu \right)$ such that $r\ne 0$ μ-almost everywhere [7]. And let Follows from theorem 5, a map ${\parallel ·\parallel }_{p:r}$ defined by is a norm on $L^p\left(\mu \right)$, which we call r-norm. And the normed space $(L^p\left(\mu \right),\parallel ·{\parallel }_{p:r})$ is called r-normed space.

Suppose an r-normed space $(L^p\left(\mu \right),\parallel ·{\parallel }_{p:r})$ with respect to a measure space $(X,\mathcal{A},\mu )$ [7]. Let $f\in L^p\left(\mu \right)$. Note that from theorem 4, we have another norm ${\parallel ·\parallel }_p:L^p\left(\mu \right)\to [0,\infty )$ defined by And follows from definition 4, we obtain

Suppose an observation which produces a finite number of data points. Let a measurable space [5] $(\Omega ,\mathcal{F})$ be the underlying event space [5]. Note that Then let $(\Omega ,\mathcal{F},P)$ be the corresponding probability space. And let $\gamma :\mathcal{F}\to \mathbb{N}$ be a counting measure [7] on $(\Omega ,\mathcal{F})$, and hence we have another measure space $(\Omega ,\mathcal{F},\gamma )$. Let $Y:\Omega \to \mathbb{R}$ be a dependent random variable such that Y has a bounded expectation [5], is almost surely nonconstant [5] and $Y\in L^2\left(\gamma \right)$. Let $r:\Omega \to \mathbb{R}$ such that And let Follows from theorem 5 and definition 4, there exists a normed space $(L^2\left(\gamma \right),\parallel ·{\parallel }_{2:r})$ such that Let $\widehat{Y}:\Omega \to \mathbb{R}$ be a predictive model on Y such that $\widehat{Y}\in L^2\left(\gamma \right)$. The model $\widehat{Y}$ is ρ-reliable if there exists some $\rho \in \mathbb{R}$ with $0<\rho <1$ such that