Types of integration So I'm studying integration as part of a real analysis module, and I've come across the regulated integral defined on regulated functions and the Riemann integral which seems to be defined for any function $f:[a,b] \rightarrow \mathbb{R}$. 
We've proved that if $f \in  R[a,b]$ (regulated function on $[a,b]$) then it is Riemann integratable and the two integrals are equal. 
I've also heard of Lebusgue integration and I'm just wondering if all these 'types' of integrals can be ordered in a sense that I've mentioned 3 'types' of integration above, so which of these are necessary given another and sufficient for another if that makes sense? Or am I going about this all wrong? And how many different 'types' of integration are there? 
If anyone has a good understanding of this topic and can understand what I'm trying to ask then it would be good if they could share some knowledge, thanks.
 A: To answer your first question all integrals over a function are equal. The integral itself is a property .Riemann integral is defined for nice functions that are continuous over an interval [a,b].
Now nice functions are the class C1 that consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable and they verify the fundamental theorem of calculus, the idea of Lebesgue integration was to verify this property for functions that are not continuous on closed intervals such as the Peano function.
Riemann integral is when you measure the area under the curve by dividing your x-axis into parts and approximation the area with a lot of rectangles, to proceed with this calculation you need to measure the length of the subdivision but this doesn't work for a large class of functions hence the idea of measure theory . Lebesgue integral is a generalization that builds from measure theory so you can verify the fundamental theorem of calculus for functions that are not continuous on a closed interval.
P.S: I am an undergraduate and have a Measure theory and Lebesgue integration course this semester, hopefully someone can verify what I said and give a better answer.
