By the layer cake representation theorem, ϕ$_{+}$ ( x ) can be written as where A + k ( f, s ) = { y ∈$_{R}$ ; f ( y ) > s + k } for any function f. The fourth equality in (A.3) follows from Fubini’s theorem. Similarly, the nonpositive function ϕ$_{−}$ ( x ) can be represented as where A − k ( f, s ) = { y ∈$_{R}$ ; f ( y ) ≤ − ( s + k ) } for any function f. Similarly, ψ$_{+}$ ( x ) and ψ$_{−}$ ( x ) are written as follows: 56 ψ$_{+}$ ( x ) = ∫ 1 0 ∞ ∑ k =0 1$_{A}$ + $_{k}$( ψ,s $_{)}$( x ) ds, ψ$_{−}$ ( x ) = − ∫ 1 0 ∞ ∑ k =0 1$_{A}$ − $_{k}$( ψ,s $_{)}$( x ) ds. ϕ$_{−}$ ( x ) = − ∫ ∞ 0 1 { ϕ$_{−}$ ( x ) ≤ − s } ds = − ∞ ∑ k =0 ∫ 1 0 1 { ϕ$_{−}$ ( x ) ≤ − ( s + k ) } ds = − ∫ 1 0 ∞ ∑ k =0 1 { ϕ$_{−}$ ( x ) ≤ − ( s + k ) } ds = − ∫ 1 0 ∞ ∑ k =0 1 { ϕ ( x ) ≤ − ( s + k ) } ds = − ∫ 1 0 ∞ ∑ k =0 1$_{A}$ − $_{k}$( ϕ,s $_{)}$( x ) ds. ϕ$_{+}$ ( x ) = ∫ ϕ$_{+}$ ( x ) 0 ds (A.3) = ∫ ∞ 0 1 { ϕ$_{+}$ ( x ) > s } ds = ∞ ∑ k =0 ∫ 1 0 1 { ϕ$_{+}$ ( x ) > s + k } ds = ∫ 1 0 ∞ ∑ k =0 1 { ϕ$_{+}$ ( x ) > s + k } ds = ∫ 1 0 ∞ ∑ k =0 1 { ϕ ( x ) > s + k } ds = ∫ 1 0 ∞ ∑ k =0 1$_{A}$ + $_{k}$( ϕ,s $_{)}$( x ) ds,