Origins of measure and integration Calculating integrals is often called as "Computing an area or a volume". More formal, integrals are measures defined in a certain space (satisfying $\sigma$-additivity and sometimes invariance conditions like translational invariance). I am asking about the very origins of measures and integrals.
Here, I think that measures and integrals arise from counting distinct "squares" or "cubes". In a multidimensional space, every Dimension is different; therefore different combinations of hypercubes can be aligned. Counting all possible combinations of hypercubes that can be aligned in a $N$-dimensional space leads to the following volume formula $V=A^N$ with the number of cubes that span one Dimension $A$. 
Is the origin of measures a combinatorial origin? Which logics lead to this concept?
 A: Can't comment yet, but I think there might be ambiguity over whether you are talking about historical origins or an abstract framework that gives the topic a place in the tree of mathematical knowledge. 
I'll assume you meant the latter.
My interpretation is that the $\sigma$-algebra gives an idea of the level of detail we want in a measure space. The simplest sigma algebras are partitions, which creates equivalence classes of points. This type of $\sigma$-algebra is very coarse in that it says that large classes of points are indistinguishable from our perspective.
Very fine $\sigma$-algebras allow us to distinguish between every point in the set. 
If you already have a topology on the set, then Borel $\sigma$-algebras allow you to base the fineness of the measure on the topology. In this case, to know the value of the measure on a particular set in the $\sigma$-algebra, one only needs to know the values on the topological sets. 
In the case that the topology is given by a metric we only have to know the value of the measure on balls of different radii. 
The case of cubes that you mentioned are products of metric spaces, and this is where it all began historically. 
The fineness of the measure becomes more important when we consider measurable functions. These are similar to measuring devices and the fineness of the $\sigma$-algebra is similar to the precision or tolerance of the measurements. An example from probability concerns two random variables $X,Y:(\Omega,\mathscr{F})\to\mathbb{R}$, where $\mathscr{F}$ is the $\sigma$-algebra on $\Omega$. One can consider the $\sigma$-algebras $\sigma(X)$ and $\sigma(Y)$ obtained by pulling back measureable sets in $\mathbb{R}$. Note that if we have given $\mathbb{R}$ a partition for a $\sigma$-algebra then $\sigma(X)$ and $\sigma(Y)$ can at most just be partitions of $\Omega$, and so won't contain much information. 
Lets give $\mathbb{R}$ the Borel $\sigma$-algebra. Conditioning $X$ on $Y$, say by considering $Z=\mathrm{E}[X|Y]$ creates a third $\sigma$-algebra. Since $Y$ already contained information, we expect that $\sigma(Z)$ should be coarser than $\sigma(X)$, and indeed, this is the case. When the space $\Omega$ consists of square integrable functions we can think of $\mathrm{E}[X|Y]$ as a projection of $L^2(\Omega,\sigma(X))$ onto a subspace $L^2(\Omega,\sigma(Y))$. 
So, as in a lot of mathematical areas, the concept of measure spaces started with a very specific set of measure spaces. At this point in time, measure was probably considered a specialized topic for people interested in integration and differential equations. Over the years the topic has been abstracted quite a bit, leading to the way that students (like me) are taught to think about it, which is much more general than its origins would suggest. 
