I have a question in my mind and let me confused however I convince my self
by a trivials answers , I would be interest to know what it does mean the
mathematicals symboles $"dx" , "dy" , dz" $ that we do
them In the RHS when we want to compute or calculate integral :
e.g: $ \int_{a}^{b}f(x)dx $ or $ \int_{a}^{b}f(x)dy $ $ \int_{a}^{b}f(x)dz $ $\cdots$
Is there someone give me analytic explanation about the reason of taking
those symboles at the end of any integral and what they do meant in mesure
theory ?
Note : I need the reason since :$dx ,dy ,dz,...$ are lebesgue mesure .
It is a historic notation. E.g. $$ f = \int \!\! df $$ was interpreted as summing (the integral symbol is a stylized "S") infinitesimal bits $df$, called a differential.
Today everyone knows that there are no infinitesimals (in $\mathbb{R}$, a few dabble in non-standard analysis), but still use this as a "Kalkül" a recipe to produce correct results most of the time. (E.g. separation of variables)
The differential indicates the integration variable. One might have come up with a different notation, that just does encode integration and the variable subject to integration. But I have experienced none so far.
In certain areas of physics the differential $df$ is written directly after the integral sign, this is useful e.g. in thermodynamics where one performs iterated integrals over many variables, where it helps to keep oversight in the formulas. It also stresses the interpretation as $\int\!\!df$ as an operator.
For the Lebesgue integral $\int f d\mu$ the $\mu$ stands for a measure. I am not sure if I ever have read or heard a term for $d\mu$, it is certainly no differential in the classical sense.