7 ON FUNCTIONAL RECORDS AND CHAMPIONS GPP V + 1. For details on GPP, see e. g. Buishand et al. (2008), Aulbach and Falk (2012 a ), Aulbach and Falk (2012 b ), and Ferreira and de Haan (2014). Proof of Proposition 2.1. Condition (5) implies that U is in the functional domain of attraction of η , see Aulbach et al. (2013, Proposition 5), i. e. Now (6) follows from Aulbach et al. (2013, Proposition 8). Next we verify (7). Choose a generator Z of ‖·‖$_{D}$ with P (sup$_{s}$$_{∈}$$_{S}$ Z$_{s}$ = c ) = 1 for some c ≥ 1. Define a measure ρ on the unit sphere ¯ C + 1 ( S ) := { g ∈ ¯ C $^{+}$( S ) : ‖ g ‖$_{∞}$ = 1 } by which is the well-known angular measure, see e. g. de Haan and Ferreira (2006, Section 9.4). By transforming to polar coordinates, we identify ¯ C $^{+}$( S ) with the product space ¯ C + 1 ( S ) × (0 , ∞ ). For the technical details of this transformation, see de Haan and Ferreira (2006, Section 9.3). On this product space, we define a product measure via dν = dρ × dr/r $^{2}$. The measure ν is well-known as the exponent measure in the literature, see again de Haan and Ferreira (2006, Section 9.3) among many others. Now having in mind that U < 1 a. s. (Hofmann (2012, Corollary 3.15)) and η < 0 a. s. (Aulbach et al. (2013, Lemma 1)), it is easy to see that (5) is equivalent with where − 1 / η is a simple MSP. Therefore, we have for all Borel sets A ⊂ ¯ C $^{+}$( S ) with ν ( ∂A ) = 0 and inf {‖ f ‖$_{∞}$ : f ∈ A } > 0, see de Haan and Ferreira (2006, Theorem 9.3.1). Define for h ∈ E ( S ) the set ν$_{n}$ ( A ) := nP ( n (1 − U )) − 1 ∈ A ) →$_{n}$$_{→∞}$ ν ( A ) 1 n max i =1 ,...,n 1 1 − U ( i ) →$_{D}$ − 1 η , ρ ( A ) := cP ( Z /c ∈ A ) , A ⊂ ¯ C + 1 ( S ) Borel , P ( n ( U − 1) ≤ f ) n →$_{n}$$_{→∞}$ P ( η ≤ f ) = exp ( − ‖ f ‖$_{D}$ ) , f ∈ ¯ E $^{−}$( S ) .