> [!theorem] > > Let $X$ be a [[Locally Compact Hausdorff Space|LCH]] space and $I \in C_0(X, \real)^*$ be a [[Bounded Linear Functional|functional]] on the continuous functions that [[Vanishes at Infinity|vanishes at infinity]]. Then there exists positive functionals $I^\pm \in C_0(X, \real)^*$ such that $I = I^+ - I^-$. > > *Proof*. Let $f \in C_0(X, [0, \infty))$. Define > $ > I^+(f) = \sup\bracs{I(g): g \in C_0(X, \real), 0 \le g \le f} > $ > then $0 \le I^+(f) \le \norm{I} \cdot \norm{f}_u$. For linearity on $C_0(X, [0, \infty))$, > 1. $I^+(cf) = cI^+(f)$ for all $c \ge 0$. > 2. If $0 \le g_1 \le f_1$ and $0 \le g_2 \le f_2$, then $0 \le g_1 + g_2 \le f_1 + f_2$, so $I^+(f_1 + f_2) \ge I(g_1) + I(g_2)$, and $I^+(f_1 + f_2) \ge I^+(f_1) + I^+(f_2)$. > 3. If $0 \le g \le f_1 + f_2$, let $g_1 = \min(g, f_1)$ and $g_2 = g - g_1$, then $0 \le g_1 \le f_1$ and $0 \le g_2 \le f_2$, so $I^+(f_1 + f_2) \le I^+(f_1) + I^+(f_2)$. > > For any $f \in C_0(X, \real)$, split $f = f^+ - f^-$, and define $I^+(f) = I^+(f^+) - I^+(f^-)$. If $f = g - h$ where $g, h \ge 0$, then $g + f^- = h + f^+$ and $I^+(g) + I^+(f^-) = I^+(h) + I^+(f^+)$. So $I^+$ does not depend on the choice of split. Thus $I^+$ is a positive linear functional on $C_0(X, \real)$, with $\norm{I^+} \le \norm{I}$. > > Let $I^- = I^+ - I$, then $I^- \in C_0(X, \real)^*$ and is positive.