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In order to store measure or M values, the geometry definition of a feature class must have the HasM property set to True (IGeometryDef::HasM = True). For shapefiles and geodatabase feature classes, you can check this property in ArcCatalog.
Once created, the geometry definition of a feature class cannot be changed. If the ‘Contains M values’ property of a feature class is No, then you will have to reload the data into a new feature class. When reloading data into a new Geodatabase feature class, you can enable M values in the output by selecting the Geometry tab and checking the 'Enable M values on the output' box.
If you are creating a new shapefile, you will have to write code to enable measure values in the output.