gistfile1.cs

public partial class Form : System.Windows.Forms.Form
{
    public Form()
    {
        InitializeComponent();
    }

    private void buttonStart_Click(object sender, EventArgs e)
    {
        StartAsyncJob();
    }

    private void StartAsyncJob()
    {
        var worker = new Worker(SynchronizationContext.Current);

        backgroundWorker.DoWork += (sender, e) => worker.DoWork();
        backgroundWorker.RunWorkerAsync();
    }
}

public class Worker
{
    private readonly SynchronizationContext _synchronizationContext;

    public Worker(SynchronizationContext synchronizationContext)
    {
        _synchronizationContext = synchronizationContext;
    }

    public void DoWork()
    {
        _synchronizationContext.Send(callback => MessageBox.Show("Hello from background worker!"), null);
    }
}