Definition (Tagged Partition)
Let \([a,b]\) be a closed interval and let \[ \mathcal{P} = \{x_0, x_1, \dots, x_n\} \] be a partition of \([a,b]\). A tagged partition of \([a,b]\) is a pair \((\mathcal{P}, \{t_1,\dots,t_n\})\) such that for each \(j = 1,\dots,n\), the point \(t_j\) satisfies \( t_j \in [x_{j-1}, x_j].\) The points \(t_j\) are called the tags of the partition.
Definition
Let \(f: [a,b] \to \mathbb{R}\). Let \(\mathcal{P}, t_j\) be a tagged partition. Then the Riemann Sum with respect to this tagged partition is the sum $$ S(f,\mathcal{P},t_j) = \sum_{j = 1}^{n} f(t_j)(x_j - x_{j-1}) = \sum_{j = 1}^{n} f(t_j) \Delta x_j $$
Definition (Riemann Integrable)
We say that \(f\) is Riemann integrable on \([a,b]\) if we can find a value \(I(f) \in \mathbb{R}\), such that for every \(\epsilon > 0\), there exists a partition \(\mathcal{P}_{\epsilon}\) such that for every partition \(\mathcal{P} \supset \mathcal{P}_{\epsilon}\) and for every corresponding tagged partition \((\mathcal{P}, t_j)\), we have $$ |S(f,\mathcal{P},t_j) - I(f)| < \epsilon $$

In other words, no matter how we choose the tags inside the subintervals, if the partition is “fine” enough, all Riemann sums will be close to the same number.

What is \(\mathcal{P}_{\epsilon}\)? \(\mathcal{P}_{\epsilon}\) is a partition depending on \(\epsilon\) such that any refinement \(\mathcal{P}\) of \(\mathcal{P}_{\epsilon}\) will produce a Riemann sum within \(\epsilon\) of the integral.

In other words, if we have this partition \(\mathcal{P}_{\epsilon}\) where every refinement (no matter what tags we choose) goes to \(I(f)\).

References