diff --git a/django/contrib/gis/gdal/geometries.py b/django/contrib/gis/gdal/geometries.py
index 6ee98c412d..44e1026e3f 100644
--- a/django/contrib/gis/gdal/geometries.py
+++ b/django/contrib/gis/gdal/geometries.py
@@ -801,14 +801,22 @@ GEO_CLASSES = {
     2001: Point,  # POINT M
     2002: LineString,  # LINESTRING M
     2003: Polygon,  # POLYGON M
+    2004: MultiPoint,  # MULTIPOINT M
+    2005: MultiLineString,  # MULTILINESTRING M
+    2006: MultiPolygon,  # MULTIPOLYGON M
+    2007: GeometryCollection,  # GEOMETRYCOLLECTION M
     3001: Point,  # POINT ZM
     3002: LineString,  # LINESTRING ZM
     3003: Polygon,  # POLYGON ZM
+    3004: MultiPoint,  # MULTIPOINT ZM
+    3005: MultiLineString,  # MULTILINESTRING ZM
+    3006: MultiPolygon,  # MULTIPOLYGON ZM
+    3007: GeometryCollection,  # GEOMETRYCOLLECTION ZM
     1 + OGRGeomType.wkb25bit: Point,  # POINT Z
     2 + OGRGeomType.wkb25bit: LineString,  # LINESTRING Z
     3 + OGRGeomType.wkb25bit: Polygon,  # POLYGON Z
-    4 + OGRGeomType.wkb25bit: MultiPoint,
-    5 + OGRGeomType.wkb25bit: MultiLineString,
-    6 + OGRGeomType.wkb25bit: MultiPolygon,
-    7 + OGRGeomType.wkb25bit: GeometryCollection,
+    4 + OGRGeomType.wkb25bit: MultiPoint,  # MULTIPOINT Z
+    5 + OGRGeomType.wkb25bit: MultiLineString,  # MULTILINESTRING Z
+    6 + OGRGeomType.wkb25bit: MultiPolygon,  # MULTIPOLYGON Z
+    7 + OGRGeomType.wkb25bit: GeometryCollection,  # GEOMETRYCOLLECTION Z
 }
diff --git a/docs/releases/5.1.txt b/docs/releases/5.1.txt
index aca1281a98..94c342e8a0 100644
--- a/docs/releases/5.1.txt
+++ b/docs/releases/5.1.txt
@@ -81,10 +81,11 @@ Minor features
 
 * :class:`~django.contrib.gis.gdal.OGRGeometry`,
   :class:`~django.contrib.gis.gdal.Point`,
-  :class:`~django.contrib.gis.gdal.LineString`, and
-  :class:`~django.contrib.gis.gdal.Polygon` now support measured geometries
-  via the new :attr:`.OGRGeometry.is_measured` and ``m`` properties, and the
-  :meth:`.OGRGeometry.set_measured` method.
+  :class:`~django.contrib.gis.gdal.LineString`,
+  :class:`~django.contrib.gis.gdal.Polygon`, and
+  :class:`~django.contrib.gis.gdal.GeometryCollection` and its subclasses now
+  support measured geometries via the new :attr:`.OGRGeometry.is_measured` and
+  ``m`` properties, and the :meth:`.OGRGeometry.set_measured` method.
 
 * :attr:`.OGRGeometry.centroid` is now available on all supported geometry
   types.
diff --git a/tests/gis_tests/gdal_tests/test_geom.py b/tests/gis_tests/gdal_tests/test_geom.py
index 35b11b753a..3967b945a4 100644
--- a/tests/gis_tests/gdal_tests/test_geom.py
+++ b/tests/gis_tests/gdal_tests/test_geom.py
@@ -675,10 +675,10 @@ class OGRGeomTest(SimpleTestCase, TestDataMixin):
             ("Point M", 2001, True),
             ("LineString M", 2002, True),
             ("Polygon M", 2003, True),
-            ("MultiPoint M", 2004, False),
-            ("MultiLineString M", 2005, False),
-            ("MultiPolygon M", 2006, False),
-            ("GeometryCollection M", 2007, False),
+            ("MultiPoint M", 2004, True),
+            ("MultiLineString M", 2005, True),
+            ("MultiPolygon M", 2006, True),
+            ("GeometryCollection M", 2007, True),
             ("CircularString M", 2008, False),
             ("CompoundCurve M", 2009, False),
             ("CurvePolygon M", 2010, False),
@@ -690,10 +690,10 @@ class OGRGeomTest(SimpleTestCase, TestDataMixin):
             ("Point ZM", 3001, True),
             ("LineString ZM", 3002, True),
             ("Polygon ZM", 3003, True),
-            ("MultiPoint ZM", 3004, False),
-            ("MultiLineString ZM", 3005, False),
-            ("MultiPolygon ZM", 3006, False),
-            ("GeometryCollection ZM", 3007, False),
+            ("MultiPoint ZM", 3004, True),
+            ("MultiLineString ZM", 3005, True),
+            ("MultiPolygon ZM", 3006, True),
+            ("GeometryCollection ZM", 3007, True),
             ("CircularString ZM", 3008, False),
             ("CompoundCurve ZM", 3009, False),
             ("CurvePolygon ZM", 3010, False),
@@ -943,6 +943,30 @@ class OGRGeomTest(SimpleTestCase, TestDataMixin):
             geom.shell.wkt, "LINEARRING (0 0 0,10 0 0,10 10 0,0 10 0,0 0 0)"
         )
 
+    def test_multi_geometries_m_dimension(self):
+        tests = [
+            "MULTIPOINT M ((10 40 10), (40 30 10), (20 20 10))",
+            "MULTIPOINT ZM ((10 40 0 10), (40 30 1 10), (20 20 1 10))",
+            "MULTILINESTRING M ((10 10 1, 20 20 2),(40 40 1, 30 30 2))",
+            "MULTILINESTRING ZM ((10 10 0 1, 20 20 0 2),(40 40 1, 30 30 0 2))",
+            (
+                "MULTIPOLYGON ZM (((30 20 1 0, 45 40 1 0, 30 20 1 0)),"
+                "((15 5 0 0, 40 10 0 0, 15 5 0 0)))"
+            ),
+            (
+                "GEOMETRYCOLLECTION M (POINT M (40 10 0),"
+                "LINESTRING M (10 10 0, 20 20 0, 10 40 0))"
+            ),
+            (
+                "GEOMETRYCOLLECTION ZM (POINT ZM (40 10 0 1),"
+                "LINESTRING ZM (10 10 1 0, 20 20 1 0, 10 40 1 0))"
+            ),
+        ]
+        for geom_input in tests:
+            with self.subTest(geom_input=geom_input):
+                geom = OGRGeometry(geom_input)
+                self.assertIs(geom.is_measured, True)
+
 
 class DeprecationTests(SimpleTestCase):
     def test_coord_setter_deprecation(self):