Add base::RunLoop and update ui_test_utils to use it to reduce flakiness

ui/views/examples/content_client/examples_browser_main_parts.cc

Index: ui/views/examples/content_client/examples_browser_main_parts.cc
diff --git a/ui/views/examples/content_client/examples_browser_main_parts.cc b/ui/views/examples/content_client/examples_browser_main_parts.cc
index c8bdba7f75e2fbfc31610abc9d1f993faca9c27e..174bc804cd90e443481671851dcfc40f99b1dde8 100644
--- a/ui/views/examples/content_client/examples_browser_main_parts.cc
+++ b/ui/views/examples/content_client/examples_browser_main_parts.cc
@@ -7,6 +7,7 @@
 #include "base/bind.h"
 #include "base/command_line.h"
 #include "base/message_loop.h"
+#include "base/run_loop.h"
 #include "base/string_number_conversions.h"
 #include "base/threading/thread.h"
 #include "base/threading/thread_restrictions.h"
@@ -77,10 +78,11 @@ bool ExamplesBrowserMainParts::MainMessageLoopRun(int* result_code) {
   // xxx: Hax here because this kills event handling.
 #if !defined(USE_AURA)
   views::AcceleratorHandler accelerator_handler;
-  MessageLoopForUI::current()->RunWithDispatcher(&accelerator_handler);
+  base::RunLoop run_loop(&accelerator_handler);
 #else
-  MessageLoopForUI::current()->Run();
+  base::RunLoop run_loop;
 #endif
+  run_loop.Run();
   return true;
 }