Measure theory and probability theory basic study notes 9 - 3.3Lp space

Lp space has been described in detail in functional analysis (but I have not learned it in detail), so here is a little more repetition.

1. Lp space definition
Let $(X,\mathcal{F},\mu )$ is a metric space, defining a function on which the power of absolute value p can be integrated ($p\ge 1$ ) The entire set is$L^p(X,F,\mu )$ . That is,$L^{}$The function in${}^p$
$${\int }_X|f{|}^{p}d\mu <\infty$$

2. Lp space is a Banach space
To demonstrate this conclusion, first demonstrate that Lp is a normed space, and then prove it completely.
To demonstrate that it is a normed space, non-negativity, positive definiteness and linearity are obviously, mainly to demonstrate the triangle inequality. Prove the triangle inequality, that is, prove the Minkowski inequality , that is:
for $1\le p<\infty ,\forall f,g\in L^p$ ,equation:
$$||f+g|{|}_p\le ||f|{|}_p+||g|{|}_p$$
To prove Minkowski's inequality, you must first prove Holder's inequality :
For a pair of conjugate numbers ( $1<p,q<\infty :\frac{1}{q}+\frac{1}{p}=1$),$\forall f,g\in L^p$ ,equation:
$$||fg||\le ||f|{|}_p+||g|{|}_q$$
And Holder's inequality depends on the following lemma :
for a pair of conjugate numbers $p,q$，对$\forall a,b\ge 0$，有：
$$a^{1/p}{b}^{1/q}\le \frac{a}{p}+\frac{b}{q}$$
The logic chain is like this, the proof is omitted here, and it can be found everywhere.

Two special p-values ​​are discussed below: $p=\infty$和$0<p<1$.
$p=\infty$ , define the norm:
∣ ∣ f ∣ ∣ ∞ = inf ⁡ { a ∈ R + , μ ( ∣ f ∣ > a ) = 0 } ||f||_{\infty}=\inf\{a\ in \R^+,\mu(|f|>a)=0\}∣∣f∣∣∞​=inf{ a∈R+,m ( ∣ f ∣>a)=0 } , it can be proved that it is indeed a norm.
$0<p<When\; 1$ , define:
$$||f|{|}_p={\int }_X|f{|}^{p}d\mu$$, note that unlike the traditional definition of the p-norm, it does not take the p-th root anymore. So obviously, it no longer satisfies the linear property,so when$0<p<At\; 1$ , it is no longer a normative space.

It can be proved that $0\le p\le \infty$ ,$L^{}$The Cauchy column in${}^{p-}$$0<p<1$ o'clock$L^p$ space is a complete distance space,$1\le p\le \infty$ is a Banach space.

3. Average convergence
Before explaining the definition of average convergence, first prove the following theorem. I think the proof of this theorem is relatively comprehensive, so it is included as follows.
Theorem Let $0<p\le \infty$ , if{ fn } ⊂ L p \{f_n\}\subset L^p{ fn​}⊂Lp satisfies:
$$\underset{n,m\to \infty }{\lim }||f_n-f_m|{|}_p=0$$
, there exists$f\in L^p$ , such that:
$$\underset{n\to \infty }{\lim }||f-f_n|{|}_p=0$$
is now called{ fn } \{f_n\}{ fn​} (order p) averagely converges to$f$ , written as:$f_n\stackrel{L_p}{\to }f$

Proof:

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Theorem: Relationship between average convergence and rest convergence
Average convergence is actually a relatively strong condition.
(1) If $f_n\stackrel{L_p}{\to }f$ ,则$f_n\stackrel{m}{\to }f$且$||f_n|{|}_p\to ||f|{|}_p$;
（2）若$f_n\stackrel{m}{\to }f$或$f_n\stackrel{to.and.}{\to }f$,则$f_n\stackrel{L_p}{\to }f$。

The proof
of (2) is more technical, and the current level is limited, only (1) is proved.