In Nowak (1989), a modular $\rho$ on a vector lattice is defined by the following properties


(N1) $\rho(x)=0\implies x=0$;


(N2) $\lvert x\rvert \le \lvert y\rvert\implies \rho(x) \le \rho(y)$;


(N3) $\rho(x\vee y )\le \rho(x)+\rho(y)$ for all $x\ge 0, y\ge 0$;


(N4) $\rho(\lambda x)\to 0$ if $\lambda \to 0$


This supposedly implies 
$$ \rho(\alpha x+\beta y)\le \rho( x)+\rho(y) \text{ for all } \alpha,\beta\ge0,\alpha+\beta=1,$$
which features in the definition of modular in Musielak and Orlicz (1959). 


**I don't see how this follows. NB: convexity is not assumed.**



Definition of an Orlicz modular space
How about this?  

Let $\alpha, \beta \ge 0$, $\alpha+\beta = 1$.  

Then
$$
|\alpha x + \beta y| \le \alpha |x| + \beta |y| \le |x| \vee |y|
$$
so
$$
\rho(\alpha x + \beta y) =
\rho\big(|\alpha x + \beta y|\big) \le \rho\big(|x| \vee |y|\big) \le 
\rho\big(|x|\big) +\rho\big(|y|\big) = \rho(x)+\rho(y)
$$


Note: $\rho(x) = \rho\big(|x|\big)$ from N2.


