Valuations on Lp-Spaces

Valuations on Banach Lattices

“…Next, we show the simpler and more convenient form that Theorem 6.1 adquires when the valuation has enough invariance with respect to measure preserving transformation. This result includes and extends the main results of [26] and [16]. Finally, we study with finer detail properties of Theorem 6.1 in the special case of L p spaces, due to their special significance.…”

“…The fact that we are free to choose X among all the order continous Banach lattices supported in Ω grants the results stated in this section a great generality. Note in particular that they imply essentially all of the main results in [26] (X = L p spaces) and [16] (X an Orlicz space) and several of the main results of [15]. 6.3.…”

“…In that context, Theorem 6.1 admits a much simpler form, Corollary 6.5. This last result includes as particular cases the main results of [26] and [16].…”

“…Natural examples of Banach lattices of functions are L p , C(K) and Orlicz spaces. Valuations on these spaces have been the object of research in [14], [15], and more recently in [26], [16], [27], [24].…”

Minkowski valuations on convex functions

“…When is a space of real valued functions, then we take to be the pointwise maximum of u and v while is the pointwise minimum. For Sobolev spaces [ 31 , 33 , 39 ] and spaces [ 34 , 46 , 47 ] complete classifications for valuations intertwining the were established. See also [ 4 , 7 , 10 , 13 , 14 , 25 , 32 , 41 , 50 ].…”