Is there alternate definitions of an integral based on values of function on open or closed interval ? In Spivak's Calculus book, he defines $\int_{a}^{b}{f}$ as the common number between all upper and lower sums, respectively, $U(f,P)$ and $L(f,P)$ for all partitions $P = \{t_0, \ldots, t_n\}$ of $[a,b]$. 
Definition of the lower sum is given as follows $L(f,P) = \sum_{i=1}^{n}m_i(t_i - t_{i-1})$, where $m_i = inf\{f(x): t_{i-1} \le x \le t_i\}$. $U(f,P)$ is defined similarly.
My Question is : 
Can I change the definition of $m_i$ to $inf\{f(x) : t_{i-1} \lt x \lt t_i\}$ to define an integral ? Does other books/author define integration this way ? 
My reason is that I am having a hard time defining integral for step functions. It will be easier for me if I do not consider the values of $f(x)$ on $\{t_0, \ldots, t_n\}$ for a partition $P$. 
 A: Your definition of the Riemann integral is equivalent to the one given in Spivak's; for the sake of the argument let's speak about Spivak-integral and hasan-integral. 
First note that an unbounded function is neither Spivak nor hasan integrable. Indeed, in both cases at least one interval will be unbounded in its interior.
To see the equivalence we can thus restrict to bounded functions, say $|f(x)|<M$ for all $x$.
For one direction, note that your lower and upper sum for a given partition are between Spivak's, hence any Spivak-integrable function is hasan-integrable and the hasan-integral equals the Spivak-integral.
For the other direction, i.e., that hasan-integrable implies Spivak-integrable, consider a partition $P=\{t_0,\ldots, t_n\}$ such that lower and upper hasan-sum differ by at most $\frac\epsilon2$ from the hasan-integral of the function. Build a new partition $P'=\{t_0, u_0, v_0, t_1, u_1, v_1, \ldots, v_{n-1},t_n\}$ where the $u_i$ and $v_i$ are chosen such that $u_i<t_i+\frac{\epsilon}{4Mn}$ and $v_i>t_{i+1}-\frac{\epsilon}{4Mn}$ (and of course also $t_i<u_i<v_{i}<t_{i+1}$). Then the contribution of the three summnds $t_i,u_i,v_i,t_{i+1}$ to the Spivak lower/upper sum for $P'$ is within $\frac\epsilon n$ of the hasan lower/upper sum for $P$, hence the total Spivak lowe/upper sums for $P'$ are within $\epsilon$ of the hasan integral. We conclude that the Spivak integral exists and has the same value.
