Measurable functions

Stein의 Measure Theory chapter 1 내용을 정리하였습니다.

1. Basic Properties

Definition 1. A function $f$ defined on a measurable subset $E$ of $\mathbb{R}^d$ is measurable if for all $a \in \mathbb{R}$ , the set

$$ f^{-1}([-\infty,a)) = \{ x \in E : f(x) <a \}$$

is measurable.

이와 관련해 몇가지 property들을 정리하겠습니다.

Property 1. The finite-valued function $f$ is measruable if and only if $f^{-1}(\mathcal{O})$ is measruable for every open set $\mathcal{O}$.

Property 2. If $f$ is continuous on $\mathbb{R}^d$, then $f$ is measurable. If $f$ is measurable and finite-valued, and $\Phi$ is continuous, then $\Phi \cdot f$ is measurable.

$\Phi$가 continous 하면 $\Phi^{-1}((-\infty,a))$는 Open set이므로, $f^{-1}\cdot \Phi^{-1}((-\infty,a))$는 measurable 하다.

Property 3. Suppose $f$ is measurable, and $ f(x) = g(x)$ for a.e. x. Then $g$ is measurable.

2. Approximation by simple functions or step functions

non-negative measurable function을 simple function 혹은 step function으로 approximation할 수 있습니다.

Theorem 4.1. Suppose $f$ is a non-negative measurable function on $\mathbb{R}^d$. Then there exists an increasing sequence of non-negative simple functions $ \{ \phi_k \}^{\infty}_{k=1}$ that converges pointwise to $f$,namely,

$$ \phi_k (x) \leq \phi_{k+1} (x),\text{and}, \lim_{k\to\infty}\phi_k (x) = f(x) , \text{for all } x.$$

Proof 

$$ F_N =  \begin{cases} f(x) & \text{if}  x \in Q_N \;\text{and}\; f(x) \leq N \\ N & \text{if}\; x\in Q_N \;\text{and}\; f(x)>N \\ 0 & \text{otherwise} \end{cases}$$

그리고 $F_N$을 $\frac{1}{M}$ 간격으로 나눕니다.

$$ E_{l,M} = \{ x\in Q_N : \frac{l}{M} < F_N(x) < \frac{l+1}{M} \} , \quad \text{for} \; 0 \leq l \leq NM $$

그리고 simple function을 정의합니다.

$$ F_{N,M} (x) =  \sum_l \frac{l}{M} \chi_{E_{l,M}} (x)$$

즉, $ 0< F_N - F_{N,M} < 1/M$ 이 됩니다.

$\varphi_k = F_{2^k,2^k}$로 정의하면, theorem에 state된 바를 얻을 수 있습니다.

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이와 마찬가지로 non-negative가 아닌, 일반적인 함수 $f$에 대해서 $ |\varphi_k | \leq |\varphi_{k+1}| $ 인 simple function들을 정의해 $f$를 approximate할 수 있습니다.

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Step function $\{ \psi_k \}$로도 $f(x)$를 근사할 수 있습니다.

Theorem 4.3 Suppose $f$ is measurable on $\mathbb{R}^d$. Then there exists a sequence of a step functions that converges pointwise to $f$ for a.e. $x$.

3.  Ergov's Theorem

Pointwise converging 하는 함수에 대해, uniform convernge하는 set을 찾을 수 있다.

Theorem 4.4. Suppose $ \{ f_k \} $ is a sequence of measurable functions defined on a measurable set $E$ with $m(E)<\infty$, and assume that $f_k \to f$ on $E$. Given $\epsilon >0$, we can find a closed set $A_{\epsilon} \subset E$ such taht $m(E-A_{\epsilon}) \leq \epsilon$ and $f_k \to f$ uniformly on $A$.