How to programmatically update datagrid

I'm trying to dynamically add records to DataGrid (which I take from mails, running in the background). The problem is that when I use the method on the other thread to start the XAML window, it just prints some records not every one and not dynamically

public AdminWindow() {
   InitializeComponent();
   foreach (Alert alert in Alert.alerts) {
      alertToDataGrid.Add(alert);
   }
   AlertTable.ItemsSource = Alert.alerts;
}

public MainWindow()
        {
            InitializeComponent();
            EmailParser parser = new EmailParser(true, "test@mail.cz", "password");
    //Could be problem with Name of the class? I had to rename it, but how I see, it didn't change it properly. But the window is showing...
        }

public EmailParser(bool isSSLuse, string username, string password) {
   this.ServerName = "imap.outlook.com";
   this.Port = 993;
   this.IsSSLuse = isSSLuse;
   this.Username = username;
   this.Password = password;
   Task.Run(() => MailKitLib(this));
}

public void MailKitLib(EmailParser emailParser) {

            bool help = true;
            Timer t = new Timer();
            t.Interval = 500;
            do
            {

                using (var client = new ImapClient())
            {
                using (var cancel = new System.Threading.CancellationTokenSource())
                {
                    client.Connect(emailParser.ServerName, emailParser.Port, emailParser.isSSLuse,
                        cancel.Token);

                    //client.AuthenticationMechanisms.Remove("XOAUTH");

                    client.Authenticate(emailParser.Username, emailParser.Password, cancel.Token);


                    var inbox = client.Inbox;
                    inbox.Open(FolderAccess.ReadOnly, cancel.Token);

                    Console.WriteLine("Total messages: {0}", inbox.Count);
                    Console.WriteLine("Recent messages: {0}", inbox.Unread);



                    for (int i = 0; i < inbox.Count; i++)
                    {
                        var message = inbox.GetMessage(i, cancel.Token);
                        Console.WriteLine("ID:" + message.MessageId);
                       
                        if (message.MessageId != null)
                        {
                            Alert alert = new Alert(message.MessageId, message.Date.DateTime, message.From.ToString(), "PING" , "LOW");                             
                            Alert.alerts.Add(alert);

                        }


                    }
                   

                    client.Disconnect(true, cancel.Token);
                }
            }
                if (t.Interval == 0)
                {
                    help = false;
                }
            } while (help != false);

Shorten the code to just this

<pre>AlertTable.ItemsSource = Alert.alerts;

System.InvalidOperationException: 'The ItemsControl object is not consistent with the source of its items.

I also tried adding a new train to contructor, I got this exception

<pre>Task.Run(() => AddItems());

System.InvalidOperationException: 'The calling thread cannot access this object because it is owned by another thread.'