Definition of double integral on a rectangle can be defined by using step functions, for example in Apostol Calculus Book. In this case the definition for step function is: * a function $s$ defined from (closed) rectangle $R=[a,b]\times [c,d]$ to the set of real number is a step function if there exists a partition P of R such that in each open sub rectangle the function $s$ is constant*.In addition, in that book, the integral of steep function on a rectangle is defined like the sum of the volume (with signal) of rectangular boxes which bases are the sub-rectangles generated by partition $P$. $$\int \int_R s(x,y)dx dy= \sum_{i=1}^{n} \sum_{j=1}^{m}c_{ij}\Delta x_i \Delta y_j$$ where $c_{ij}$ is the constant value of $s$ in the open sub-rectangle $ (x_{i-1}, x_i) \times (y_{j-1}, y_j)$.
My questions are: (a) Is it necessary that $s$ function to be bounded in that definition? (b) If answer to my before question is "not", how can I proof that a step function (like I will give below) is integrable on $R$ when we use Riemann Sums definition like in Apostol Analysis Book?
I am thinking for example in the function $$s(x,y)=\begin{cases} 1/x & \text{ if } y=0 \text{ and } x>0 \\ 2 & \text{ in other case }. \end{cases} $$
under definition above $s$ is an integrable step function (its integral is 2), but how can I give the same answer by using definition using Riemann Sums?
I have read on the sets of Zero Content, but theorem related to the existence of the integral assumes that function $s$ is bounded on $R$.