Uniqueness of simple unsigned integral

Question

This is a uniqueness problem of the Simple unsigned integral that I'm struggling with.

A function $f$ is called simple unsigned function if it can be written as $$f(x)=\sum_{i=1}^k c_i 1_{E_i}(x)$$ for some $k \in \mathbb{N}$, $c_i \in [0,+\infty] ~\forall~ i=1(1)k~$ where $E_i$'s are Lebesgue measurable sets.

The simple unsigned integral of the function $f$ is defined as $$Simp \int_{\mathbb{R}^d}f(x)dx:=\sum_{i=1}^k c_im(E_i)$$ where $m(\cdot)$ denotes the Lebesgue measure.

The problem is to show that the simple unsigned integral $f \to Simp \int_{\mathbb{R}^d}f(x)dx$ is the only map from the space Simp$^+(\mathbb{R}^d)$ to $[0,+\infty]$ that obeys all the following properties:

$(i)$ (Unsigned linearity) Let $\alpha, \beta \in [0, +\infty]$. $$Simp \int_{\mathbb{R}^d}\{\alpha f(x)+\beta g(x)\}dx=\alpha \times Simp \int_{\mathbb{R}^d}f(x)dx+\beta \times Simp \int_{\mathbb{R}^d}g(x)dx$$

$(ii)$ (Finiteness) $Simp \int_{\mathbb{R}^d}f(x)dx < +\infty$ iff $f$ is finite a.e. and the support of $f$ is finite.

$(iii)$ (Vanishing) $Simp \int_{\mathbb{R}^d}f(x)dx = 0$ iff $f=0$ a.e.

$(iv)$ (Equivalence) $f=g$ a.e. $\implies$ $Simp \int_{\mathbb{R}^d}f(x)dx=Simp \int_{\mathbb{R}^d}g(x)dx$

$(v)$ (Monotonicity) $f \leq g$ a.e. $\implies$ $Simp \int_{\mathbb{R}^d}f(x)dx \leq Simp \int_{\mathbb{R}^d}g(x)dx$

$(vi)$ (Compatibility with Lebesgue measure) For any Lebesgue measurable set $E$, $$Simp \int_{\mathbb{R}^d}1_E(x)dx=m(E)$$

I'm defining an arbitrary functional $M:$ Simp$^+(\mathbb{R}^d) \to [0,+\infty]$ that follows the properties and I do realize that I have to show the that $$M(f)=k \times Simp \int_{\mathbb{R}^d}f(x)dx$$ using the properties $(i)-(v)$ and then to use property $(vi)$ (Compatibility with Lebesgue measure) to get the normalizing constant $k=1$. But I'm not being able to penetrate the first part. Any help would be greatly appreciated!

Answer 1

Suppose that $M : \mathbf{Simp}^+(\mathbb R^d) \to [0,\infty]$ is a map satisfying (i) and (vi).

Let $f \in \mathbf{Simp}^+(\mathbb R^d)$, say $f = \sum_{i=1}^{k}c_i 1_{E_i}$.

By (i), $M(f) = \sum_{i=1}^{k}c_i M(1_{E_i})$.

By (vi), $M(1_{E_i}) = m(E_i)$.

Therefore, $$M(f) = \sum_{i=1}^{k} c_i m(E_i) = \operatorname{Simp}\int_{\mathbb R^d}f(x)\ dx$$

So, we only need properties (i) and (vi) to establish uniqueness. Now you just need to show that the simple unsigned integral does indeed satisfy (ii) through (v).