I presume you mean "proper convexity' as in $(1)$ above not $(2)$

Not just mere, " "midpoint convex/jensen convexity"/ "convexity in the sense of Jensen"," as in $(2)$, below.

Although (its hardly mere I suppose) because the $(1)$ as defined in the question and $(2)$ below are, not, but are nonetheless "almost equivalent".

(2) $$F(\frac{x+y}{2}) \leq \frac{F(x)}{2} +\frac{F(y)}{2}$$

That is because under relatively mild conditions measurability, regularity conditions/boundedness conditions, midpoint convex function (in the sense of Jensen) are convex in the tradition .

Apparently a real valued midpoint convex function $(2)$ already satisfies the definition of convexity as above $(1)$, except for the restriction that $\sigma$ applies only to all rational numbers in the the unit interval (not just $2$, or dy-adics). That is before continuity is applied.

That is $(1.a)$ below according to pt. 7.11 of chapter "Continuous Convex Functions" in http://link.springer.com/chapter/10.1007%2F978-3-7643-8749-5_7

$$(1.a)\forall \, \sigma \in \mathbb{Q}\cap[0,1];\, \forall (x,y)\, \in\, \text{dom}(F):\, F(\,\sigma x + [1-\sigma] y\,)\, \leq\, \sigma F(x)\,+\,[1-\sigma]F(y).$$

Its presumably a bit confusing the use the words 'in the sense of Jensen' for midpoint convexity.

This is because I believe that Jensen did, or helped develop the inequalities that any 'function' must satisfy, in order to be 'properly convex',as defined above in the question $(1)$.

Because Jensen well as the weaker notion of midpoint convexity $(2)$ which apparently is equivalent to $(1.a)$, at least if the domain is real valued, and which is named after Jensen (midpt convexity is often called jensen- convexity).