I assume that you are speaking of a MPLS L3VPN. Most service providers will not run an IGP directly with their customer. Some will but then you trust that your customer is capable of handling that IGP and you need to make sure that what goes into the RIB from the customers IGP never comes back into your own IGP. If it does it could break the whole SP network because even though the customer is in a VRF the separation is not as clear as with BGP.
The benefit of an IGP is that it converges fast compared to BGP.
BGP gives a clearer separation as it meant to be used for interdomain routing. With BGP it's also easier to do aggregation, traffic policy and using the same policy for entire AS. In short it was designed for exactly these scenarios while IGP was designed to be run inside the network.
Daniel Dib
CCIE #37149
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Daniel Dib
CCIE #37149
CCDE #20160011
Please rate helpful posts.