There are a few caveats to integrating vuex-module-decorators with nuxt. The first steps are described in the README (although I missed it because it nuxt was in the small text).

In addition to this, according to issue #179, there are a few other caveats

• The file name has to be the same as the module name
• The store should go in ~/store, not ~/store/modules
• You have to export default your module classes