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I agree with the principles introduced in this motion However I find the formulation of exclusion of infinity too abstract to be useful For example it does not provide a way to evaluate a expresssion such that f( (-\infty, a] ) As a tentative solution to this difficulty, I propose the following approach The number set under consideration is R The basic intervals have the general form [a,b] where a and b are reals numbers such that a<= b At least two order relations can be defined over basic intervals The usual inclusion and rthe <=_I relation defined by <=_i ={([a,b], [c,d] ) | a <= c and b <= d } Only this last relation structure the basic interval set as a lattice This lattice is incomplete. To insure the completeness one has to had to add the following intervals (-infty,a] = inf_x {[x,a] } [b,+infty] = sup_y { [b,y] } (-infty, +infty) = sup_y{ (-infty, y]) = inf_x ({[x, +infty)}) (-infty,-infty) is the infinum of the whole interval set ( basic intervals + extensions) (+infty,+infty) is the supremum of the whole interval set ( basic intervals + extensions) In such a context the value of f( (-infty,a] ) is defined only as the limit for any sequence of f([x,a]) when [x,a] goes to (-infty, a] It is undefined if such a limit does not exist. As an illustration we consider the sine fonction Of course the calculation of sin( (-infty,1] is problematic due to the difficulties of evaluating corerectly sin( (x,1]) However we ca note that clearly range (f, (-infty, 1]) = [-1, 1] despite the fact that sin(-infty) cannot be defined . Conversely on has clearly 1/[0,1] = [1,+infty) I hope this can help Best regards Dominique LOHEZ . Wolff v. Gudenberg a écrit : Dear members of the working group. -- Dr Dominique LOHEZ ISEN 41, Bd Vauban F59046 LILLE France Phone : +33 (0)3 20 30 40 71 Email: Dominique.Lohez@xxxxxxx |