We live in an era where, through the detection of gravitational waves, the discovery of their first electromagnetic counterparts, and with the full maturation of neutrino astronomy, astrophysics has entered a new phase; the “classic photon” astronomy must therefore integrate into a broader framework of “multi-messenger” astronomy. Multi-messenger astronomy is, in fact, a “time-domain astronomy”, in which the classic pointing of know celestial objects with small-field instruments is accompanied and supported by instruments, such as LSST in the optical band, capable of observing or monitoring the entire sky. In this new scientific perspective, it is imperative for X-ray astronomy to contribute through the use of new instruments that can overcome the limitations of traditional space observatories. Global angular coverage of the sky is fundamental and extremely difficult to achieve with instruments on board satellites due to their necessarily limited resources. The lunar site offers new opportunities to overcome these limitations, allowing for the deployment of an X-ray observatory capable of observing half of the sky simultaneously, but also allowing access to the entire solid angle by exploiting the rotation of the moon around its own axis. The Lunar Electromagnetic Monitor in X-rays (LEM-X) is an All Sky Monitor for the X-ray band (2-50 keV) based on the concept of coded aperture camera. The basic element is the camera of the Wide Field Monitor, currently under study for the WFM experiment on the sino-European eXTP mission. Each camera is equipped with four Large Area Silicon Drift Detectors developed in Italy by a collaboration of INFN-INAF-ASI-FBK and produced in the foundries of the Bruno Kessler Foundation in Trento. The advantages of this design are: high sensitivity, high level of maturity, compact and modular design, and high level of redundancy. The concept is to realize a "dome" composed of N identical modules that observe different directions, thus covering an overall simultaneous field of view of 2π. Each module is autonomous, compact (about 35x35x35 cm3), light (~10 kg), and with low power consumption (15 W). Moreover, thanks to its modular nature, LEM-X can be built gradually -even over a long period of time- and can eventually reach dimensions, weight, and complexity impossible to achieve on a free-flyer. The real breakthrough of LEM-X is represented by the simultaneous coverage of half of the sky in an energy band, that of “soft” X-rays, still not covered by wide-field experiments in orbit, with imaging and spectroscopic capabilities. The lunar site, as part of a wider infrastructure, can prove ideal for an all sky monitoring experiment. The stability of the site allows for full exploitation of the instrument's capabilities, obtaining high-resolution images of transient events and detected sources. If positioned appropriately (e.g. along the lunar equator), LEM-X will have simultaneous access to half of the celestial sphere and periodically to the entire sky. The site's accessibility will allow for maintenance and updates over time, making LEM-X a long-term observatory infrastructure, similar to a ground-based telescope. The base module of the experiment is currently under development within the eXTP collaboration, in cooperation with Spain, Germany, Poland, France, Denmark, and Turkey. Due to time constraints, the experiment concept mentioned here has not yet been discussed within that collaboration, but if a study were to proceed, it would be natural to propose to our European colleagues with whom we collaborate on eXTP to also contribute to LEM-X.