Agree with Chris W. This field calculator expression will calculate from and to measures in a point file, providing there is a single river - polyline:
mxd = arcpy.mapping.MapDocument("CURRENT")
with arcpy.da.SearchCursor(lr, "Shape@") as cursor:
for row in cursor:
Run it on a field(double) in the original points table using
to calculate FROM measure. To compute TO measure, replace 0 by 1.
Create route from river, create events (line). The rest according to Michael solution.