You must be missing a namespace. You could do something like this:
MainWindow.cs:
 using System;
 using System.Text;
 using System.Windows;
 using System.Windows.Controls;
namespace YourNameSpaceHere
{
    public partial class MainWindow : Window
    {
        internal static string message_ = string.Empty;
        MainWindow()
        {
            InitializeComponent();
            SetupMessage();
        }
        private void SetupMessage()
        {
            message_ = "Hello World!";
        }
    }
}
OtherFile.cs :
using System;
using System.Text;
using System.Windows;
using System.Windows.Controls; //are you missing this namespace?
namespace YourNameSpaceHere
{
        public class Messaging
        {
            Messaging()
            {
                MessageBox.Show(MainWindow.message_);
            }
        }
}
Note that I am using the same namespace in each file and by using the internal keyword the message can be accessed by anything as long as it is in the same namespace. I hope this helps!