wpfeap.cs

private void button_Click(object sender, RoutedEventArgs e)
{
    // ここは UI スレッドで実行される
    Debug.WriteLine("button_Click : {0}", Thread.CurrentThread.ManagedThreadId);

    var client = new WebClient();
    client.Encoding = Encoding.UTF8;
    client.DownloadStringCompleted += this.OnDownloadStringCompleted;
    client.DownloadStringAsync(new Uri("http://tech.blog.aerie.jp"));
}

private void OnDownloadStringCompleted(object sender, DownloadStringCompletedEventArgs e)
{
    // ここも UI スレッドで実行されている
    Debug.WriteLine("OnGetResponseCompleted : {0}", Thread.CurrentThread.ManagedThreadId);

    this.webBrowser.NavigateToString(e.Result);
}