A dart board has no left and right to slice along, yet its expected score is easy to find: sort the board into the regions that score each value, then add value times area. That is the Lebesgue integral of a simple function, and it is how expectations over price paths are computed, where no left-to-right order of the outcomes exists. Throw darts and watch the average score settle on the sorted sum. Switch the thrower from uniform to aimed: every region keeps its score, only the measure changes, and the expectation changes with it.