I have taken out a book to help me understand measure theory better, one of the examples (that one has to do alone) asks:
'Show there is a closed subset of [0,1] that contains no interval but it's Lebesgue measure exceeds 0.999'
I've had a good look at questions similar to this one and have got some general ideas of how to solve this problem however I'm very unsure how to go about it: Let us call our subset A, if λ(A)>0.999 with λ being the Lebesgue measure then we would have to consider [0,1] as the union of intervals [0,0.001]⋃[0.001,0.002]⋃...⋃[0.999,1] and deduce A from here however we've already said that A is closed so using rationals wouldn't work as [0,1]∩Q isn't closed in R, maybe that means constructing R\Q however the example has already stated A contains no intervals.
I'm very confused, how do I go about finding this result?