Yes, that it is a set expression. And since it is so, 'half' does not mean 'the larger part' at all: it is idiomatic.
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Mister Micawber 'half' does not mean 'the larger part' at all: it is idiomatic.Well, here is how my Longman displays the entry half (1) [predeterminer, pronoun, adjective]
MUSCOVITE 2 most of (sic!) the largest part of something.That's only Longman's opinion.
Mister MicawberI never think of the meaning as 'most' at allI don't either. I think it's closer to "a great deal of", but less than half. For this reason it strikes me as an indicator of speaker exaggeration.