Vitali-Carathéodory Theorem
Theorem
Let $\struct {X, \tau}$ be a locally compact Hausdorff space.
Let $\MM$ be a $\sigma$-algebra over $X$ which contains the Borel $\sigma$-algebra generated by $\tau$.
Let $\mu$ be a Radon measure on $\MM$.
Let $f \in \map {\LL^1} \mu$, where $\map {\LL^1} \mu$ denotes the (real) Lebesgue 1-space of $\mu$.
For all $\epsilon \in \R_{>0}$, there exists some $\tuple {u, v} \in \paren {X^\R}^2$ such that:
- $u$ is upper semicontinuous and bounded above
- $v$ is lower semicontinuous and bounded below
- $u \le f \le v$
and :
- $\ds \int_X \paren {v - u} \rd \mu < \epsilon$.
Proof
Let:
- $\forall x \in X: \map f x \ge 0$
and:
- $\exists x \in X: \map f x \ne 0$
By Measurable Function is Pointwise Limit of Simple Functions, there exists a sequence:
- $\sequence {s_n} \in \paren {\map \EE \MM}^\N$
where $\map \EE \MM$ denotes the space of simple functions on $\struct {X, \MM}$.
By Pointwise Difference of Simple Functions is Simple Function, the differences of consecutive terms in a sequence of simple functions are simple functions.
By Limit of Sequence is Sum of Difference of Consecutive Terms, there exists a sequence $\sequence {t_n}$ of simple functions such that:
- $\ds f = \sum_{i \mathop = 1}^\infty t_n$
By the definition of simple functions, each simple function is a finite linear combination of characteristic functions.
Thus there exists some $\tuple {\sequence {E_i}, \sequence {c_i} } \in \MM^\N \times \R_{>0}^\N$ such that:
- $\ds f = \sum_{i \mathop = 1}^\infty c_i \chi_{E_i}$
Now:
\(\ds \sum_{i \mathop = 1}^\infty c_i \map \mu {E_i}\) | \(=\) | \(\ds \sum_{i \mathop = 1}^\infty \int_X c_i \chi_{E_i} \rd \mu\) | Integral of Characteristic Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_X \sum_{i \mathop = 1}^\infty c_i \chi_{E_i} \rd \mu\) | Monotone Convergence Theorem (Measure Theory) | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_X f \rd \mu\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \infty\) | as $f \in \map {\LL^1} \mu$ |
That is, the series:
- $\ds \sum_{i \mathop = 1}^\infty c_i \map \mu {E_i}$
Denote by $\powerset X$ the power set of X.
By the definition of Radon measure, for all $\epsilon \in \R_{>0}$, there exists some $\tuple {\sequence {K_i}, \sequence {V_i} } \in \paren {\paren {\powerset X}^\N}^2$ such that for all $i \in \N$:
and
- $c_i \map \mu {V_i - K_i} < 2^{-\paren {i + 1} } \epsilon$
By Characteristic Function of Open Set is Lower Semicontinuous:
- for all $i \in \N$, $\chi_{V_i}$ is lower semicontinuous.
By Constant Multiple of Lower Semicontinuous Function is Lower Semicontinuous:
- for all $i \in \N$, $c_i \chi_{V_i}$ is lower semicontinuous.
Define:
- $\ds v = \sum_{i \mathop = 1}^\infty c_i \chi_{V_i}$
By Series of Lower Semicontinuous Functions is Lower Semicontinuous:
- $v$ is lower semicontinuous.
By Characteristic Function of Compact Set is Upper Semicontinuous:
- for all $i \in \N$, $\chi_{K_i}$ is upper semicontinuous.
By Constant Multiple of Upper Semicontinuous Function is Upper Semicontinuous:
- for all $i \in \N$, $c_i \chi_{K_i}$ is upper semicontinuous.
By definition of convergent series, for all there exists some $N \in \N$ such that:
- $\ds \sum_{i \mathop = N + 1}^\infty c_i \map \mu {E_i} < \frac \epsilon 2$
Define:
- $\ds u = \sum_{i \mathop = 1}^N c_i \chi_{K_i}$
By Finite Sum of Upper Semicontinuous Functions is Upper Semicontinuous, $u$ is upper semicontinuous.
Now, for all $i \in \N$:
- $\chi_{K_i} \le \chi_{E_i} \le \chi_{V_i}$
So:
- $u \le f \le v$
Now:
\(\ds \int_X \paren {v - u} \rd \mu\) | \(=\) | \(\ds \int_X \paren {\sum_{i \mathop = 1}^N c_i \paren {\chi_{V_i} - \chi_{K_i} } + \sum_{i \mathop = N + 1}^\infty c_i \chi_{V_i} } \rd \mu\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \int_X \paren {\sum_{i \mathop = 1}^\infty c_i \paren {\chi_{V_i} - \chi_{K_i} } + \sum_{i \mathop = N + 1}^\infty c_i \chi_{E_i} } \rd \mu\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^\infty \int_X c_i \paren {\chi_{V_i} - \chi_{K_i} } \rd \mu + \sum_{i \mathop = N + 1}^\infty \int_X c_i \chi_{E_i} \rd \mu\) | Monotone Convergence Theorem (Measure Theory) | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^\infty c_i \map \mu {V_i - K_i} + \sum_{i \mathop = 1}^\infty \map \mu {E_i}\) | Integral of Characteristic Function: Corollary | |||||||||||
\(\ds \) | \(<\) | \(\ds \sum_{i \mathop = 1}^\infty \frac \epsilon {2^{i + 1} } + \frac \epsilon 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \epsilon\) |
$\blacksquare$
Source of Name
This entry was named for Giuseppe Vitali and Constantin Carathéodory.