Greatest Lower Bound of the Set of Upper Bounds for a Function

Question

I'm currently in the process of reading a paper, and am trying to work through some of the details on my own. In order to ask my question more effectively, I'm going to begin with a little background:

def: A modular on a set $X$ is a function $w: (0,\infty) \times X \times X \longrightarrow [0, \infty]$ satisfying the following $\forall x,y,z \in X$:

1. $x=y$ iff $w(\lambda, x, y)=0$, $\forall \lambda >0$.
2. $w(\lambda, x, y) = w(\lambda, y, x)$, $\forall \lambda >0$.
3. $w(\lambda + \mu, x, y) \leq w(\lambda, x, z)+ w(\mu, y, z)$, $\forall \lambda, \mu >0$.

Now, the paper claims at one point that given $x_{0} \in X$, the set $X_{w} = \left\{x\in X: \lim_{\lambda \to \infty} w(\lambda, x, x_{0})=0\right\}$ is a metric space with metric $d_{w}^{\circ}(x,y) = \inf \left\{ \lambda > 0 : w(\lambda, x, y)\leq \lambda\right\}$, called a modular space.

I am attempting to show the metric space axioms for the given metric, and my proofs for both $0 \leq d_{w}^{\circ}(x,y)<\infty$ $\forall x,y$ (I'll refer to this henceforth as Property I),

and the subadditivity property $\displaystyle d_{w}^{\circ}(x,z) \leq d_{w}^{\circ}(x,y)+d(y,z)$, $\forall x,y,z$ (I'll refer to this henceforth as Property IV) depend on the following assumption:

$\sup w(\lambda,x,y) \leq \{\lambda >0 : w(\lambda, x, y) \leq \lambda\}$, since $ \{\lambda >0 : w(\lambda, x, y) \leq \lambda\}$ is the set of all upper bounds for $w$. And in fact, it is the greatest lower bound for this set, so we have that $\mathbf{\sup w(\lambda, x, y) = \inf \{ \lambda > 0 :w(\lambda, x, y) \leq \lambda \}}$.

My question is whether this assumption is true.

Using it, I am able to prove easily the finiteness of $w$ in Property I, and prove Property IV as follows:

Let $\eta = \lambda + \mu$. Then, by definition of a modular, $w(\eta, x, y) \leq w(\lambda,x,z)+w(\mu, y, z)$ $\forall \lambda, \mu > 0$.

Thus,

$\sup w(\eta, x, z) \leq \inf\left[ w(\lambda, x, y) + w(\mu, y, z)\right] \leq \sup\left[ w(\lambda, x, y) + w(\mu, y, z)\right] \leq \sup \left[w(\lambda, x, y)\right] + \sup \left[ w(\mu, y, z)\right]$, which, by my assumption, implies that

$\inf\left\{\eta > 0: w(\eta, x, z) \leq \eta \right\} \leq \inf\left\{\lambda > 0: w(\lambda, x, z) \leq \lambda \right\} + \inf\left\{\mu > 0: w(\mu, x, z) \leq \mu \right\}$,

and thus that $\displaystyle d_{w}^{\circ}(x,z) \leq d_{w}^{\circ}(x,y)+d(y,z)$, $\forall x,y,z$.

Please let me know if my assumption is correct; without it, my proofs of Properties I and IV go completely out the window.

And, if it's not correct, how would I go about proving these two properties?

Thank you ahead of time.

Answer 1

Your logic is quite hard to follow, but you answer your own question with:

$\sup\;w(λ,x,y)≤\{λ>0:w(λ,x,y)≤λ\}$, since $\{λ>0:w(λ,x,y)≤λ\}$ is the set of all upper bounds for $w$. And in fact, it is the greatest lower bound for this set, so we have that $\sup\;w(λ,x,y)=\inf\{λ>0:w(λ,x,y)≤λ\}$.

Reading this, you claim:

$\{λ>0:w(λ,x,y)≤λ\}$ is the set of all upper bounds for $w$.

All means every, and so by definition must include the greatest lower bound.