Linear functionals which share the properties of the integral In my multi-dimensional real analysis class, we have recently defined the definite integral over $\mathbb{R}$ as so:

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be continuous. Define the support of $f$, $supp(f)$, to be the closure of the set $\lbrace x \in \mathbb{R}^n:f(x) \neq0 \rbrace$. If $supp(f)$ is compact, we write $f\in C_{00}(\mathbb{R^n})$.
Let $\int_{a}^{b} f(x) \mathrm{d}x$ denote the familiar Riemann integral. If $f\in C_{00}(\mathbb{R})$, then there exist real numbers $a$,$b$ such that $supp(f)\subseteq [a,b]$. Let $$\int_{\mathbb{R}} f(x) \mathrm{d}x=\int_{\mathbb{R}} f =\int_{a}^{b} f(x) \mathrm{d}x$$ for any such $a$, $b$.

We then showed that:

$\int_{\mathbb{R}} (f+g)=\int_{\mathbb{R}} f+\int_{\mathbb{R}} g$ and $\int_{\mathbb{R}} \alpha f=\alpha\int_{\mathbb{R}} f$ for $\alpha \in \mathbb{R}$. This proves that $$\int_{\mathbb{R}}: C_{00}(\mathbb{R}) \rightarrow \mathbb{R}$$
is a linear functional

We haven't really discussed what a "functional" is, but I assume it is a function which acts on other functions.
Finally, we showed that:

(1) If $f(x) \geq0$ for all $x \in \mathbb{R}$, then $\int_{\mathbb{R}} f \geq 0$, (2) $|\int_{\mathbb{R}} f| \leq \int_{\mathbb{R}} |f|$, and (3) $\int_{\mathbb{R}} f(x-c)\mathrm{d}x=\int_{\mathbb{R}} f(x)\mathrm{d}x$ for $c \in \mathbb{R}$.

According to my lecturer, the definite integral is the only linear functional which also satisfies properties (1), (2), and (3), up to a multiplicative constant. I was surprised at this, and am wondering what background one needs to prove this; I assume functional analysis is needed (which I can't take yet).
 A: $C_{00}(\mathbb{R}^{n})$ is a linear space under natural operations: The sum of two functions is defined as $(f+g)(x)=f(x)+g(x)$ and the scalar multiple of a function is defined as $(\alpha f)(x)=\alpha f(x)$. With these operations you have a linear space. A linear functional on this space is a linear function $F : C_{00}(\mathbb{R}^{n})\rightarrow\mathbb{R}$. "Functional" is a word reserve for a (linear) function on a linear space whose values are in the scalar field of the linear space. The integral is a functional because it takes functions and returns a scalar value, which is the integral of the function.
I suppose you could consider the result one of Functional Analysis, but I would consider the result one of measure theory, too. That's a mix. The only translation invariant, order preserving, positive linear functional is--up to a positive constant--integration with respect to Lebesgue measure.
Where Functional Analysis comes into play is showing that the only continuous linear functionals on $C_{00}(\mathbb{R}^{n})$ are related to integration with respect to signed regular measures.
