21 ON FUNCTIONAL RECORDS AND CHAMPIONS But elementary computations show that Δ b $_{a}$H$_{1}$ = 0, i.e., Q is the null measure on ( − ∞ , 0 ]. Instead one can define Q on [ −∞ , 0] $^{d}$\ {− ∞ } by putting for x$_{i}$ ≤ 0 and i = 1 , . . . , d Then Q has its complete mass on the set { ⋃ d i =1 ( {−∞} i − $^{1}$× ( −∞ , 0] ×{−∞} d − i )} and This approach is closely related to the formulation of the exponent measure theorem as in Balkema and Resnick (1977) and Vatan (1985). Take, on the other hand, ‖·‖$_{D}$ = ‖·‖$_{∞}$ , which is the least D -norm. In this case, the components η$_{1}$, . . . , η$_{d}$ of η are completely dependent, i.e., η$_{1}$ = η$_{2}$ = · · · = η$_{d}$ a.s. and, thus, which is a max-stable distribution (MSD). H$_{∞}$ ( x ) = E (∥ ∥ (min( x$_{i}$, η$_{1}$ )) d i =1 ∥ ∞ ) − ‖ x ‖$_{∞}$ = E (max( ‖ x ‖$_{∞}$ , η$_{1}$ )) − ‖ x ‖$_{∞}$ = exp( − ‖ x ‖$_{∞}$ ) , x = ( x$_{1}$, . . . , x$_{d}$ ) ≤ 0 ∈ R $^{d}$, Q ( × d i $_{=1}$[ −∞ , x$_{i}$ ] \ {− ∞ } ) = Q ( d ⋃ i =1 ( {−∞} × · · · × {−∞} × ( −∞ , x$_{i}$ ] × {−∞} × · · · × {−∞} ) = d ∑ i =1 Q ( {−∞} × · · · × {−∞} × ( −∞ , x$_{i}$ ] × {−∞} × · · · × {−∞} ) = 1 d d ∑ i =1 exp( x$_{i}$ ) . Q ( {−∞} × · · · × {−∞} × ( −∞ , x$_{i}$ ] × {−∞} × · · · × {−∞} ) := exp( x$_{i}$ ) d .