xml - How to iterate through XML in Powershell?

Question

I have this XML document in a text file:

<?xml version="1.0"?>
<Objects>
  <Object Type="System.Management.Automation.PSCustomObject">
    <Property Name="DisplayName" Type="System.String">SQL Server (MSSQLSERVER)</Property>
    <Property Name="ServiceState" Type="Microsoft.SqlServer.Management.Smo.Wmi.ServiceState">Running</Property>
  </Object>
  <Object Type="System.Management.Automation.PSCustomObject">
    <Property Name="DisplayName" Type="System.String">SQL Server Agent (MSSQLSERVER)</Property>
    <Property Name="ServiceState" Type="Microsoft.SqlServer.Management.Smo.Wmi.ServiceState">Stopped</Property>
  </Object>
</Objects>

I want to iterate through each object and find the DisplayName and ServiceState. How would I do that? I've tried all kinds of combinations and am struggling to work it out.

I'm doing this to get the XML into a variable:

[xml]$priorServiceStates = Get-Content $serviceStatePath;

where $serviceStatePath is the xml file name shown above. I then thought I could do something like:

foreach ($obj in $priorServiceStates.Objects.Object)
{
    if($obj.ServiceState -eq "Running")
    {
        $obj.DisplayName;
    }
}

And in this example I would want a string outputted with SQL Server (MSSQLSERVER)