Dealing with collection of simple or step functions as a function space Can someone show that the collection of all simple functions is a function space? Also, is the collection an algebra?
Thank you very much for your help...
 A: If I understand right:
A function space is a vector space $W$ of real-valued functions on some set $X$ with scalars in $\mathbb{R}$(since you're taking min, max), with the additional properties that
$\min(f,g) \in W$ and also $\max(f,g) \in W$.
We must simply need to show that $\max(\sum_{i=1}^{L1} r_i \chi_{I_i}, \sum_{j=1}^{L2} \rho_j \chi_{J_j}) = \sum_{k=1}^{L3} p_k \chi_{K_k}$. Well, we can even assume that the $I_i$'s are disjoint(if not, we can split up the sum further into a finite sum of characteristic functions on disjoint sets), so assume they are disjoint. then let $L3=L1 \cdot L2$, and define for $ 1 \leq m \leq L3$( with $m = qj + t$ for $0 \leq q < L1$ and $1 \leq t \leq L2$) the interval $K_m=I_q \cap I_t$ and define $p_k = \max(r_i,\rho_j)$, similarly to define $\min(\sum_{i=1}^{L1} r_i \chi_{I_i}, \sum_{j=1}^{L2} \rho_j \chi_{J_j}) = \sum_{k=1}^{L3} p_k \chi_{K_k}$ we would simply define $p_k = \min(r_i,\rho_j)$.
This should be a more or less trivial verification away, pick a random point and verify this construction gives the correct value of $\max(f,g)$ and $\min(f,g)$
What do you mean by algebra, an algebra over a field(vector space with bilinear product)? If so yes, just take pointwise multiplication.
