diff --git a/CHANGES.rst b/CHANGES.rst
index 461f2af6c..fd26bc0f6 100644
--- a/CHANGES.rst
+++ b/CHANGES.rst
@@ -23,7 +23,11 @@ upgrading your version of coverage.py.
 Unreleased
 ----------
 
-Nothing yet.
+- Fix: setting ``COVERAGE_CORE=sysmon`` no longer errors on 3.11 and lower,
+  thanks `Hugo van Kemenade <pull 1747_>`_.  It now issues a warning that
+  sys.monitoring is not available and falls back to the default core instead.
+
+.. _pull 1747: https://github.com/nedbat/coveragepy/pull/1747
 
 
 .. scriv-start-here
diff --git a/coverage/collector.py b/coverage/collector.py
index 65e329822..87cd620e3 100644
--- a/coverage/collector.py
+++ b/coverage/collector.py
@@ -152,6 +152,7 @@ def __init__(
             core = os.getenv("COVERAGE_CORE")
 
             if core == "sysmon" and not env.PYBEHAVIOR.pep669:
+                self.warn("sys.monitoring isn't available, using default core", slug="no-sysmon")
                 core = None
 
             if not core:
diff --git a/doc/cmd.rst b/doc/cmd.rst
index a440e987c..439927d9c 100644
--- a/doc/cmd.rst
+++ b/doc/cmd.rst
@@ -273,6 +273,11 @@ Conflicting dynamic contexts (dynamic-conflict)
   :meth:`.Coverage.switch_context` function to change the context. Only one of
   these mechanisms should be in use at a time.
 
+sys.monitoring isn't available, using default core (no-sysmon)
+  You requested to use the sys.monitoring measurement core, but are running on
+  Python 3.11 or lower where it isn't available.  A default core will be used
+  instead.
+
 Individual warnings can be disabled with the :ref:`disable_warnings
 <config_run_disable_warnings>` configuration setting.  To silence "No data was
 collected," add this to your configuration file:
diff --git a/tests/test_process.py b/tests/test_process.py
index 8146e0d17..43518066e 100644
--- a/tests/test_process.py
+++ b/tests/test_process.py
@@ -1129,10 +1129,13 @@ def test_core_request_sysmon(self) -> None:
         out = self.run_command("coverage run --debug=sys numbers.py")
         assert out.endswith("123 456\n")
         core = re_line(r" core:", out).strip()
+        warns = re_lines(r"CoverageWarning: sys.monitoring isn't available", out)
         if env.PYBEHAVIOR.pep669:
             assert core == "core: SysMonitor"
+            assert not warns
         else:
             assert core in ("core: CTracer", "core: PyTracer")
+            assert warns
 
 
 class FailUnderNoFilesTest(CoverageTest):