Consider a real value function $ P $ defined by parts in a domain $ A $ as follows: $ A $ is divided into non-overlapping subsets $ A_1, ldots, A_n $and there is a function $ P_i: A_i rightarrow mathbb {R} $ defined in each region. The general function $ P $ is defined by $ P (x) = P_i (x) $ when $ x in P_i $, which specifies exclusively $ P $ throughout $ A $.

(Please note I want $ P $ ALWAYS have an average between the infimum and the supremum of $ P $range of.)

Now consider $ lambda $ like the Lebesgue measure. Leave $ A = bigcup A_i subseteq mathbb {R} $. You want a finite additive invariant measure of translation $ m $ in $ mathbb {R} $ so that

## Case 1

$ m (A cap (a, b)) = lambda (A cap (a, b)) $ when $ A cap (a, b) $ It has a full measure of lebesgue.

## Case 2

$ m (A cap (a, b)) = b-a $ when $ A $ it is dense in $ (a, b) $; Y, $ m (A_1 cap (a, b)) = b-a $ Yes $ A_1 $ is the only dense partitioned subset in $ (a, b) $.

## Case 3

Yes $ lambda (A cap (a, b)) $ it is between $ 0 $ Y $ b-a $; Y, $ A $ it's finite then $ m (A cap (a, b)) = lambda (A cap (a, b)) $

Yes $ lambda (A cap (a, b)) = 0 $ Y $ A $ it is dense as a finite set $ T $ in $ (a, b) $; what if $ A_1 $ it is dense as a finite set $ T_1 $, $ A_2 $ it is dense as a finite set $ T_2 $, etc. then yes $ bigcup T_i subseteq T $

$ m (A_i cap (a, b)) = left | T_i cap (a, b) right | / left | T cap (a, b) right | $

## Case 5

Yes $ lambda (A cap (a, b)) = 0 $ Y $ A $ is finite then $ m (A_i cap (a, b)) = left | A_i cap (a, b) right | / left | A cap (a, b) right | $

## Case 6

Yes $ A cap (a, b) $ is positive but not full, so it should (somehow) partition $ (a, b) $ in subintervals where one of the first four conditions MUST apply for each subinterval. (If not, see case 6).

If the first condition applies to any of the subintervals, take the measure of those subintervals and give the remaining subintervals the measure zero. If the first condition does not apply to any subinterval; but, the second yes, takes the measure of those subintervals and gives the remaining subintervals the measure zero. If the first and second conditions do not apply to any subinterval; but, the third one does, it takes the measure of those subintervals and gives the remaining subintervals the zero measure. If the first, second, and third conditions do not apply to any of the subintervals, and the fourth condition do, take the measure of those subintervals and give the remaining subintervals the measure zero.

## Case 6

Any case that does not match the previous cases can have a measure between 0 and the length of the subinterval where case 6 is kept.

However, we must have a measure where the first six cases are true.

If such a measure exists, use it to find the "average" value of some function $ P $ defined in each $ A_i $.

Before saying that such a measure does not exist (remember that it is finitely additive, it does not have to be accountingly additive), read the following:

EXTENSIVE DENSITIES AND MEASURES IN GROUPS AND SPACES G

AND ITS COMBINATORY APPLICATIONS

SOLECKI'S SUB-MEASURES AND DENSITIES IN GROUPS

DENSITIES, MEASURES AND GROUP PARTITIONS