Gathering data, monitoring performance and assessing metrics in order to facilitate service improvements are all activities associated with which part of the service lifecycle?

Service Operation encompasses the day-to-day activities of monitoring services, collecting operational data, and assessing performance metrics through processes such as Event Management and operational reporting. This operational data is then fed into Continual Service Improvement to drive enhancements, but the actual gathering and monitoring activities are executed within Service Operation. It is the phase where services are 'run' and therefore where live performance information is collected.

Capacity Management is a process - not a service lifecycle phase - and while it involves some performance data, it cannot be the answer to a question asking which part of the lifecycle these activities belong to.

Service Design focuses on designing new or changed services and does not perform live monitoring or data gathering on services already in production.

Availability Management is a process, not a lifecycle phase, and its scope is narrower than the broad data gathering and metrics assessment activities described in the question.