Vitali-Carathéodory Theorem

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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:

$\ds \sum_{i \mathop = 1}^\infty c_i \map \mu {E_i}$

converges.

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 $i \in \N$:

$K_i$ is compact
$V_i$ is open
$K_i \subset E_i\subset V_i$

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.