How to verify a SSAS Backup and how to perform Integrity Checks in SSAS?
I found a post by Darren Gosbell, a SQL Server MVP (at the time at least).  His post can be found here:

http://geekswithblogs.net/darrengosbell/archive/2008/09/13/ssas-verifying-backups.aspx

He points out that you cannot run this from your desktop, but you can run it on the server to do the verification.  The instructions from 2008 are:

Navigate to:   &lt;Program Files&gt;\Microsoft SQL Server\MSAS10.SQL08\OLAP\bin 

Then type:  msmdsrv /? 


Look at the commands available with msmdsrv and see what you want to do.  (I am away from a SSAS installation at the moment, so I cannot verify, but it should be worth checking out.)

EDIT:
2) How to run a DBCC CheckDB in SSAS? (database integrity)

Starting with SQL 2016 we have a DBCC CHECKDB command for SSAS for Dimensional as well as Tabular


  DBCC provides on-demand database validation for Multidimensional and
  Tabular databases on an Analysis Services instance. You can execute
  DBCC in an MDX or XMLA query window in SQL Server Management Studio
  (SSMS) and trace the DBCC output in either SQL Server Profiler or
  xEvent sessions in SSMS. The command takes an object definition and
  returns either an empty result set or detailed error information if
  the object is corrupted. In this article, you’ll learn how to run the
  command, interpret results, and address any problems that arise.