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:

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: