Motivation:
So, I had an idle thought last week, and I thought I would ask it here before I forget about it. It is well known that we can define $$ e^x = \lim_{n \to \infty} \left(1 + \frac x{n} \right)^{n} $$ Where $x$ here can be taken as either a number or a linear operator.
This is often intuitively explained as stating that $e^x$ is the multiplication which is generated by the infinitessimal perturbation of $1$ in the direction of $x$. Or, if you prefer, $e^x$ is the "continuous interest rate" that is generated by the periodic interest rate $x$.
In either case, it is "suspicious" that we've broken the $n$th product down into $n$ identical pieces, so perhaps we can come up with a more "robust" definition. In that vein:
Problem Statement
Let $\lambda$ denote a tuple $(\lambda_1,\lambda_2,\dots,\lambda_n)$. Let $\Lambda$ denote the set of all such (finite) tuples of positive $\lambda_i$ satisfying $\sum_i\lambda_i = 1$ with the partial order $$ (\lambda_{1,1},\dots,\lambda_{1,k_1},\lambda_{2,1},\dots,\lambda_{2,k_2}, \dots \dots \dots,\lambda_{n,1},\dots,\lambda_{n,k_n}) \succeq\\ ([\lambda_{1,1}+\cdots+\lambda_{1,k_1}],[\lambda_{2,1}+\cdots+\lambda_{2,k_2}], \dots,[\lambda_{n,1}+\cdots+\lambda_{n,k_n}]) $$ So, for example, $(1/2,1/2) \preceq (1/4,1/4,1/2) \preceq (1/8,1/8,1/4,1/2)$.
Define the net $(e_{\lambda}^x)_{\lambda \in \Lambda}$ by $$ e_{\lambda}^x = \prod_{i =1}^n \left( 1 + \lambda_i x\right) $$
Conjecture: $\lim_{\lambda \in \Lambda} e_\lambda^x = e^x$
Is this statement correct? Has this been done before? Is this demonstrably useless? Let me know.