Add test for build_owner.0.dart (#157499)

Contributes to https://github.com/flutter/flutter/issues/130459

It adds a test for
- `examples/api/lib/widgets/framework/build_owner.0.dart`

dev/bots/check_code_samples.dart

@@ -317,7 +317,6 @@ final Set<String> _knownMissingTests = <String>{
   'examples/api/test/widgets/navigator/restorable_route_future.0_test.dart',
   'examples/api/test/widgets/navigator/navigator_state.restorable_push.0_test.dart',
   'examples/api/test/widgets/focus_manager/focus_node.unfocus.0_test.dart',
-  'examples/api/test/widgets/framework/build_owner.0_test.dart',
   'examples/api/test/widgets/nested_scroll_view/nested_scroll_view_state.0_test.dart',
   'examples/api/test/widgets/scroll_position/scroll_metrics_notification.0_test.dart',
   'examples/api/test/widgets/media_query/media_query_data.system_gesture_insets.0_test.dart',

examples/api/lib/widgets/framework/build_owner.0.dart

@@ -8,12 +8,33 @@ import 'package:flutter/rendering.dart';
 /// Flutter code sample for [BuildOwner].
 
 void main() {
-  WidgetsFlutterBinding.ensureInitialized();
-  final Size size = measureWidget(const SizedBox(width: 640, height: 480));
-
-  // Just displays the size calculated above.
   runApp(
-    WidgetsApp(
+    const BuildOwnerExample(),
+  );
+}
+
+class BuildOwnerExample extends StatefulWidget {
+  const BuildOwnerExample({
+    super.key,
+  });
+
+  @override
+  State<BuildOwnerExample> createState() => _BuildOwnerExampleState();
+}
+
+class _BuildOwnerExampleState extends State<BuildOwnerExample> {
+  late final Size size;
+
+  @override
+  void initState() {
+    super.initState();
+    size = measureWidget(const SizedBox(width: 640, height: 480));
+  }
+
+  @override
+  Widget build(BuildContext context) {
+    // Just displays the size calculated above.
+    return WidgetsApp(
       title: 'BuildOwner Sample',
       color: const Color(0xff000000),
       builder: (BuildContext context, Widget? child) {
@@ -23,8 +44,8 @@ void main() {
           ),
         );
       },
-    ),
-  );
+    );
+  }
 }
 
 Size measureWidget(Widget widget) {

examples/api/test/widgets/framework/build_owner.0_test.dart

@@ -0,0 +1,22 @@
+// Copyright 2014 The Flutter Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style license that can be
+// found in the LICENSE file.
+
+import 'package:flutter/material.dart';
+import 'package:flutter_api_samples/widgets/framework/build_owner.0.dart' as example;
+import 'package:flutter_test/flutter_test.dart';
+
+void main() {
+  testWidgets('BuildOwnerExample displays the measured size', (WidgetTester tester) async {
+    await tester.pumpWidget(const example.BuildOwnerExample());
+
+    expect(find.text('Size(640.0, 480.0)'), findsOne);
+  });
+
+  test('The size of the widget is measured', () {
+    expect(
+      example.measureWidget(const SizedBox(width: 234, height: 567)),
+      const Size(234, 567),
+    );
+  });
+}