how to check the filesystem type of a logical volume

Same as you would with any other block device. e.g.

file -s /dev/vg1/lv1

If it's ext4, it'll say something like:

/dev/vg1/lv1: Linux rev 1.0 ext4 filesystem data, UUID=xxxx, volume name "yyyy" (needs journal recovery) (extents) (large files) (huge files)

Alternatively, you could run blkid /dev/vg1/lv1. That would report something like:

/dev/vg1/lv1: LABEL="yyyy" UUID="xxxx" TYPE="ext4"

From man file:

-s, --special-files

Normally, file only attempts to read and determine the type of argument files which stat(2) reports are ordinary files. This prevents problems, because reading special files may have peculiar consequences. Specifying the -s option causes file to also read argument files which are block or character special files.

This is useful for determining the filesystem types of the data in raw disk partitions, which are block special files. This option also causes file to disregard the file size as reported by stat(2) since on some systems it reports a zero size for raw disk partitions.

Use lsblk -f.

$ lsblk -f
    NAME            FSTYPE      LABEL          UUID                                   FSAVAIL FSUSE% MOUNTPOINT
    sda                                                                                              
    ├─sda1          ext4                       d2123fec-6c94-426f-b505-8cf3441122cf    742,6M    17% /boot
    └─sda2          LVM2_member                BLt1F3-S5E3-hBRX-Eqya-AUOx-wyxc-a5Yaf4                
      ├─fedora-root xfs                        59935050-8e89-4f7f-bf0f-2f448c4c680b     12,8G    15% /
      └─fedora-swap swap                       5fbca8c7-fd22-4acf-a411-d96a100c8ab2                  [SWAP]
    sr0             iso9660     VBox_GAs_6.0.8 2019-05-13-13-58-35-65