On main chain. Via lightning you can support all the capacity of Visa/Mastercard/banks and then some. Main chain provides the security for lightning, lightning provides the transaction storage space and infrastructure.
The lightning infrastructure, if you graph it, looks very similar to existing global payment networks. The difference is that transactions settle instantly because they are protected by the underlying blockchain and they are automated with no middlemen to delay things. No complicated currency conversions, no banks negotiating liquidity in blocks manually and having to buy/sell other assets to stay in balance, no bank holidays, less fees. Which means you can take your money from person-to-person faster, which reduces friction in the economy. Which is exactly what a good currency should be.