You want samtools depth.

Here's one way:

$ bedmap --echo --count --echo-overlap-size --echo-ref-size --delim '\t' my_intervals.bed <(bam2bed < my_reads.bam) | awk -vFS="\t" -vOFS="\t" '{ print $0, ($NF-1)/$NF; }' > answer.bed

Note: Please see my second answer in this thread, which uses your specific inputs.

First, please sort the BED file with sort-bed (not sortBed or UNIX sort):

$ sort-bed 10A_AT1_SE1.bed > 10A_AT1_SE1.sorted.bed

Then map the sorted BED file to the BAM file:

$ bedmap --skip-unmapped --delim '\t' --echo --count --bases-uniq --echo-ref-size --bases-uniq-f 10A_AT1_SE1.sorted.bed <(bam2bed < 10A_1_H3K.sorted.bam) > 10A_1_H3K.coverage.bed

The file 10A_1_H3K.coverage.bed will look something like this:

$ head 10A_1_H3K.coverage.bed
chr1    199064  200374  404 1310    1310    1.000000
chr1    6853080 6870040 204 9044    16960   0.533255
chr1    8004008 8027135 507 22308   23127   0.964587
chr1    8061658 8098833 366 37175   37175   1.000000
chr1    8174215 8211962 369 34253   37747   0.907436
chr1    8873384 8890703 434 17319   17319   1.000000
chr1    8874708 8890922 420 16214   16214   1.000000
chr1    15833039    15850691    271 17652   17652   1.000000
chr1    16643023    16645543    266 2520    2520    1.000000
chr1    16665212    16667822    281 2610    2610    1.000000

The last four columns are:

• The number of reads in the BAM file that overlap an interval in the BED file by one or more bases
• The number of bases in the interval in the BED file that have coverage by reads from the BAM file
• The length of the interval in the BED file
• The fraction of bases in the interval in the BED file that have coverage by reads from the BAM file

The order of the specified bedmap operands --count, --bases-uniq, --echo-ref-size, and --bases-uniq-f write the aforementioned column values in that same order.

If one base of read coverage is too lenient, additional operands are available to further constrain that criterion. Run bedmap --help and take a look at the "Overlap Options" section.

Hope this helps!