Is there more than one Lebesgue Integral? I am studying Measure Theory, and am a bit confused about the way that the Lebesgue Integral is defined.
Wikipedia states that:

We denote the integral of an extended Borel function $f$ as following:
$$ \int _{\Omega} f d\mu = \int _{\Omega} f^+ d\mu - \int _{\Omega}f^- d\mu $$

But here there is seemingly no constraint on what our choice of Measure $\mu$ is.
Whereas, I have previously been led to believe that the Lebesgue Integral specifically refers to integration with respect to the Lebesgue Measure $m$ on $\mathbb{R}$.
I would be interested in knowing which of these these is correct or if I am possibly misunderstanding something. And if Lebesgue integration is the more general claim of integration with respect to any measure, then does this not mean that we have multiple different Lebesgue integrals depending on our choice of measure?
 A: We define the (Lebesgue) integral with respect to an arbitrary measure space $(X,\mathcal{M},\mu)$, where $X$ is a set, $\mathcal{M}$ is a $\sigma$-algebra of subsets of $X$, and $\mu$ is an arbitrary measure. If we let $(X,\mathcal{M},\mu)=(\mathbb{R}^n,\mathcal{L},m)$, where $m$ is the Lebesgue measure and $\mathcal{L}$ is the Lebesgue measurable subsets of $\mathbb{R}^n$, we get one possible integral, but we could just as easily take $\mu=\mu_c$, the counting measure on $\mathbb{R}^n$ (where $\mu(A)$ is equal to the number of points in $A$), and $\mathcal{M}=2^X$, so that $\int_A f d\mu_c= \sum_{x_a \in A}f(x_a)$.
In the context of measure theory, we usually just use the term "integral" to refer to the integral defined above, and talk about the "integral with respect to the Lebesgue measure" when talking about the special case mentioned in your question, though this could also just be called the "integral" when the measure space is clear from context.
Edit: One more thing, the definition you give is the way you get from the integral defined for positive functions to the more general definition, and that isn't really important for your question. The definition you should make sure you understand is the one involving the suprema of integrals of simple functions; the definition you give follows very naturally after you understand this.
