Try setting the mode of the binding to TwoWay
:
<tool:ListPicker Name="txt_LZZ"
ItemsSource="{Binding ZZR}" SelectedItem="{Binding MySelectedItem, Mode=TwoWay}" />
I also wouldn't instantiate your value collection in the getter, but instead make the setter private, and assign a value to the property in your view model (e.g. the constructor or when the view model is activated).