Metamath Proof Explorer


Theorem mblvol

Description: The volume of a measurable set is the same as its outer volume. (Contributed by Mario Carneiro, 17-Mar-2014)

Ref Expression
Assertion mblvol ( 𝐴 ∈ dom vol → ( vol ‘ 𝐴 ) = ( vol* ‘ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 volres vol = ( vol* ↾ dom vol )
2 1 fveq1i ( vol ‘ 𝐴 ) = ( ( vol* ↾ dom vol ) ‘ 𝐴 )
3 fvres ( 𝐴 ∈ dom vol → ( ( vol* ↾ dom vol ) ‘ 𝐴 ) = ( vol* ‘ 𝐴 ) )
4 2 3 syl5eq ( 𝐴 ∈ dom vol → ( vol ‘ 𝐴 ) = ( vol* ‘ 𝐴 ) )