My background is physics and engineering, not specifically mathematics though, of course, I learned some advanced mathematics for my profession. So, please forgive me if my terminology could be more technically precise or correct, and I welcome suggestions on how to make it so.
A Riemann integral of a function, $f$, over an interval, $[0,1]$, of the real-valued $x$ axis results in the partition approaching being countably infinite, as the upper limit (stopping point; an integer) of $\Sigma$ approaches infinity . This means that not every real number in the interval becomes a member of the partition. Also, $f$ is only evaluated at a number of points in the interval that is countably infinite, and is not evaluated at every point in the interval.
Is there a way to generalize the concept of integral so that every real number in the interval of integration is included in the partition? That would also result in $f$ being evaluated at every real value of $x$ in the interval. I have no idea how to do this, but just wondering if it can be done and if it would converge even for well behaved test functions.
The reason I am asking is motivated by my experience in physics. For a discrete physical variable, $k$, you can define $\sum_{k=1}^{n} f(k)$ where $f$ is evaluated at every point in the discrete interval $[1,n]$. $f$ can be represented as a vector if the range of k is finite. If we generalize $k$ to a continuous variable, $x$, then the generalization of the sum is an integral, $\int_{1}^{n} f(x) dx$, but $f$ is no longer evaluated at every value of $x$ in the interval of integration. So, I was wondering if there was a way to define integral such that for both the sum of a function of a discrete variable or the integral of a function of a continuous variable, the function, $f$, is evaluated at every value of the variable in the interval of interest. Of course, I would want the result of the integral, defined in this way, to be the same as it is when defined in the traditional way, since we know that the traditional way yields physically meaningful results.
This is just a kind of pure thought question. I am just wondering, that is all. There is no specific need for what I am asking about.